REGULARITY RESULTS
FOR HAMILTON-JACOBI EQUATIONS
Ph.D. Thesis
Supervisor
Candidate
Prof. Stefano Bianchini
Daniela Tonon
ACADEMIC YEAR 2010/2011
Il presente lavoro costituisce la tesi presentata da Daniela Tonon, sotto
la direzione di ricerca del prof. Stefano Bianchini, al fine di ottenere
l’attestato di ricerca post-universitaria di Doctor Philosophiae in Analisi
Matematica presso la SISSA di Trieste. Ai sensi dell’art. 18, comma 3,
dello Statuto della SISSA pubblicato sulla G.U. n. 62 del 15.03.2001, il
predetto attestato è equipollente al titolo di Dottore di Ricerca in Matematica.
Trieste, anno accademico 2010/2011.
Contents
Introduction
0.1 Hamilton-Jacobi equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.2 Decomposition for BV functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.3 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
7
14
16
1 Preliminaries
1.1 Rectifiable sets . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Hausdorff distance and Hausdorff convergence of sets . . . . .
1.3 Generalized differentials . . . . . . . . . . . . . . . . . . . . .
1.4 Decomposition of a Radon measure . . . . . . . . . . . . . . .
1.5 BV and SBV functions . . . . . . . . . . . . . . . . . . . . . .
1.6 Semiconcave functions . . . . . . . . . . . . . . . . . . . . . .
1.7 Viscosity solutions . . . . . . . . . . . . . . . . . . . . . . . .
1.8 Properties of viscosity solutions when H = H(t, x, p) . . . . .
1.8.1 Minimizers and Generalized Backward Characteristics
1.8.2 Backward solutions . . . . . . . . . . . . . . . . . . . .
1.9 Properties of viscosity solutions when H = H(p) . . . . . . .
1.9.1 Duality solutions . . . . . . . . . . . . . . . . . . . . .
2
SBV Regularity for Hamilton-Jacobi equations
2.1 No-crossing property for a small interval of time
2.2 Proof of Theorem 2.1 . . . . . . . . . . . . . . . .
2.2.1 Preliminary remarks . . . . . . . . . . . .
2.2.2 Construction of the functional F . . . . .
2.2.3 Hille-Yosida transformation . . . . . . . .
2.2.4 Approximation and area estimates . . . .
2.2.5 Conclusion of the proof . . . . . . . . . .
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31
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3 Jacobian’s regularity
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4 SBV-like regularity for Hamilton-Jacobi equations: the
4.1 Extension and preliminary properties of the vector field d
4.2 General strategy . . . . . . . . . . . . . . . . . . . . . . .
4.3 One-dimensional case . . . . . . . . . . . . . . . . . . . . .
5
convex case
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55
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58
60
Contents
4.4
The multi-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Some applications
5.1 Generalized Hydrostatic Boussinesq equations .
5.1.1 One-dimensional case . . . . . . . . . . .
5.2 Sticky particles . . . . . . . . . . . . . . . . . . .
5.3 Convex solutions of Hamilton-Jacobi equations in
5.4 Multi-dimensional case . . . . . . . . . . . . . . .
5.4.1 A counterexample . . . . . . . . . . . . .
6 Decomposition of BV
6.1 The Decomposition
6.2 The Decomposition
6.3 Counterexamples .
62
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functions
Theorem for Lipschitz functions from Rn to R . . . . . . . .
Theorem for BV functions from Rn to R . . . . . . . . . . .
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the multi-dimensional case
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Acknowledgements
105
Bibliography
107
6
Introduction
The most part of this thesis is devoted to the study of the regularity of viscosity solutions of
Hamilton-Jacobi equations under different hypotheses on the Hamiltonian.
In the last part of this thesis we present a decomposition theorem for BV functions, which
extends the Jordan decomposition property.
0.1
Hamilton-Jacobi equations
In this thesis we consider the Hamilton-Jacobi equation
∂t u + H(t, x, Dx u) = 0
in Ω ⊂ [0, T ] × Rn ,
(1)
where H : R × Rn × Rn is called Hamiltonian and Ω is an open domain in Rn+1 .
It is well known that, even under strong regularity assumptions on the Hamiltonian H and
on the initial datum for the Cauchy problem related to (1), classical solutions exist only for a
finite interval of time: indeed solutions of the Cauchy problem develop discontinuities of the
gradient. The reason why Hamilton-Jacobi equations don’t have in general smooth solutions for
all times can be explained by the method of characteristics, see Evans [26]. However if we try to
overcome the problem considering solutions which satisfy the equation only almost everywhere
uniqueness is lost. The introduction of viscosity solutions, see Crandall and Lions [23], Crandall,
Evans and Lions [22], and Lions [32], solves the problem of existence, uniqueness and stability
even in the case of a continuous initial datum.
In general viscosity solutions are only locally Lipschitz continuous. The structure of the
non differentiability set of viscosity solutions has been studied by Fleming [28], Cannarsa and
Soner [21] and others. As a major assumption these authors restrict the problem to the case
where the Hamiltonian H(t, x, p) is convex with respect to p and smooth in all variables. Under this restriction the viscosity solution of (1) can be represented as the value function of a
classical problem in Calculus of Variation, see Fleming and Rishel [29]. Indeed, adding suitable
assumptions, the viscosity solution is precisely the value function
{
}
∫ t
˙
u(t, x) := min u0 (ξ(0)) +
L(s, ξ(s), ξ(s))ds ξ(t) = x, ξ Lipschitz continuous in [0, t] ,
0
where L, the Lagrangian, is the Legendre transform of H with respect to the last variable and
u0 is the initial datum at time t = 0. Moreover, when the Hamiltonian is strictly convex, the
viscosity solution is locally semiconcave in both the variables, see Cannarsa and Sinestrari [20].
7
Introduction
In particular, for every K ⊂⊂ Ω, there is a constant C > 0 such that the function (t, x) 7→
u(t, x) − C(t2 + |x|2 ) is concave on K. The semiconcavity of u ensures that Du = (∂t u, Dx u)
is BVloc , in particular u is twice differentiable almost everywhere and its distributional Hessian
is a matric of measures with locally bounded variation. Intuitively one can figure out viscosity
solutions as functions which are Lipschitz and whose gradient is piecewise smooth, undergoing
jump discontinuities along a family of surfaces of codimension one (in space and time).
Deeper results on fine regularity properties of viscosity solutions have been proved. Using
geometric measure theory and the classical method of characteristics, Fleming proved, in [28],
that viscosity solutions inherit the regularity of the initial datum in the complement of the closed
set Λ ∪ Γ, Λ being the non differentiability set of u and Γ the set of conjugate points for the
variational problem associated to (1). A significant result, which confirms the intuitive picture
given above, was obtained by Cannarsa, Mennucci and Sinestrari in [19]. Requiring the strict
convexity of H in the last variable, they proved the SBV regularity of the gradient Du, when u is
the viscosity solution of the Cauchy problem of (1) with a regular initial datum u(0, x) = u0 (x)
belonging to W 1,∞ (Rn ) ∩ C R+1 (Rn ), with R ≥ 1. Furthermore they give a sharper estimate on
the set of regular conjugate points Γ \ Λ, which implies that this set has Hausdorff dimension
at most n − 1 if the initial datum is C ∞ . In particular they proved that the closure of the non
differentiability set, which is equal to Λ ∪ Γ, is Hn -rectifiable. However the techniques used to
obtain these results do not apply when the initial datum is less regular.
The question on the existence of such an SBV-regularizing effect for the gradient of a viscosity
solution of the Hamilton-Jacobi equation was addressed by several other authors. For a recent
survey on the topic see De Lellis [24]. The motivation for studying this kind of regularity arises
from problems in Control Theory, in image segmentation and measure-theoretic questions.
A result, which confirms this SBV-regularizing effect, has been proved by Ambrosio and De
Lellis, see [4], for the entropy solution U of the scalar conservation law
∂t U + Dx (H(U )) = 0
in Ω ⊂ [0, T ] × R.
(2)
Theorem 0.1 (Ambrosio and De Lellis). Let U ∈ L∞ (Ω) be an entropy solution of (2) with
H ∈ C 2 (R) and locally uniformly convex. Then, there exists S ⊂ [0, T ] at most countable such
that ∀t ∈ [0, T ] \ S the following holds:
U (t, ·) ∈ SBVloc (Ωt )
with Ωt := {x ∈ R| (t, x) ∈ Ω}.
In particular U ∈ SBVloc (Ω).
Thanks to the equivalence between the entropy solution U and the gradient of a viscosity
solution of a related Hamilton-Jacobi equation, which holds in the one-dimensional case, the
same result applies to Dx u when u is a solution of the Hamilton-Jacobi equation
in Ω ⊂ [0, T ] × R.
∂t u + H(Dx u) = 0
This equivalence is in general not true in the multi-dimensional case.
A recent generalization of the SBV-regularizing effect to the multi-dimensional case has been
proved by Bianchini, De Lellis and Robyr in [9].
8
Introduction
Theorem 0.2 (Bianchini, De Lellis and Robyr). Let u be a viscosity solution of
∂t u + H(Dx u) = 0,
(3)
in Ω ⊂ [0, T ] × Rn , assume H belongs to C 2 (Rn ) and
c−1
H Idn (p) ≤ Hpp (p) ≤ cH Idn (p)
for some cH > 0. Then the set of times
S := {t| Dx u(t, ·) ∈
/ SBVloc (Ωt )}
is at most countable. In particular Dx u ∈ [SBVloc (Ω)]n , ∂t u ∈ SBVloc (Ω).
Due to this result, when the Hamiltonian is uniformly convex, the singular part of the matrix
of Radon measures Dx2 u is concentrated on a Hn−1 -rectifiable set, the measure theoretic jump set
J of Dx u. This prevents the Hessian of u from having a Cantor part. Analogous considerations
hold for Dx ∂t u and ∂t2 u.
The paper of Ambrosio and De Lellis and the one of Bianchini, De Lellis and Robyr prove
the SBV-regularizing effect using a strategy whose idea originates from a conjecture pointed
out by Bressan, during a conversation on the problem with De Lellis. If Dx u(t̄, ·) is not SBV
for a certain time t̄, then at future times t̄ + ε the Cantor part of Dx2 u(t̄, ·) gets transformed
into jump singularities. Roughly speaking this allows to conclude that Dx u(t, ·) is SBV out
of a countable number of t’s and that Dx u is SBV as a function of two variables. Motivated
by this conjecture, the general idea of the proof consists in constructing a monotone bounded
functional F (t), whose jumps are related to the presence of a Cantor part in |Dx2 u(t, ·)|. Since
the boundedness and the monotonicity of this functional imply that it can have only a countable
number of jumps, the Cantor part of |Dx2 u(t, ·)| can be different from zero only for a countable
number of t’s. A key role is played by the map
Xt,0 (x) := x − tHp (Dx+ u(t, x)),
where Dx+ u(t, x) is the superdifferential of the semiconcave function u(t, ·), and by its restriction
χt,0 (·) to the set Ut where Dx+ u(t, x) is single-valued. Indeed, the functional F (t) measures
exactly the area of a set transported along characteristics from time t to time 0
F (t) := Hn (χt,0 (Ut )).
Thus the properties of characteristics are important. Since H depends only on p, characteristics
are straight lines and are called optimal rays. Their no-crossing property ensures the injectivity
of the map χt,0 . Other useful ingredients for the proof are two estimates on the measure of
the area of a set transported along optimal rays. The first one ensures that, for any Borel set
A ⊂ Ωt ,
∫
Hn (Xt,0 (A)) ≥ c1 Hn (A) − c2 t
d(∆u(t, x)),
A
9
Introduction
where c1 , c2 are positive constants and ∆u(t, x) is the Laplacian of u(t, ·), which is a Radon
measure. The second ensures that, for any Borel set A ⊂ Ωt and 0 < δ < t,
(
)
t−δ n n
n
H (Xt,δ (A)) ≥
H (Xt,0 (A)).
t
The no-crossing property and the two estimates are enough to prove that, when the Cantor part
of |Dx2 u(t, ·)|(Ωt ) is positive, then there exists a set A ⊂ Ut of null Hn -measure over which the
Cantor part is concentrated and such that
χt,0 (A) ∩ χt+δ,0 (Ut+δ ) = ∅
for all δ > 0. Therefore the presence of a Cantor part in |Dx2 u(t, ·)| corresponds to a jump for
F (t). The geometrical theory of monotone functions, see for example Alberti and Ambrosio [1],
plays a crucial role in handling the details for the multi-dimensional case.
Theorem 0.2 can be seen also as a kind of generalization of the result of Cannarsa, Mennucci and Sinestrari in [19]. Indeed, in the case of H = H(p) uniformly convex, Theorem 0.2
contains part of that result since it proves SBV regularity reducing the regularity of the initial
datum to bounded Lipschitz functions. Nothing is said, however, about the closure of the non
differentiability set.
The result of Bianchini, De Lellis and Robyr suggested to us different directions of research.
i) A first question is about the preservation of the SBV-regularizing effect in the case of a
Hamiltonian dependending on space and time, H(t, x, p), which is uniformly convex in
the last variable and a bounded Lipschitz initial datum. As already seen the result of
Cannarsa, Mennucci and Sinestrari [19] applies to this kind of Hamiltonians when they
are strictly convex in the last variable (a slightly weaker requirement) and in the case of a
regular initial datum (a stronger requirement). Therefore a positive answer could be seen
as a kind of generalization of this result. Moreover it will extend the result of Bianchini,
De Lellis and Robyr [9] to the case of a general Hamiltonian depending also on time and
space’s variables.
ii) Theorem 0.2 implies that the Jacobian J(t, ·) := Hn (Dx+ u(t, ·)) is a measure which has
only an absolute continuous part with respect to Hn and a part which is concentrated on
a Hn−1 -rectifiable set and is absolute continuous with respect to Hn−1 . One can wonder if
this measure has only integer parts, i.e. parts which are concentrated on a Hk -rectifiable
set and are absolute continuous with respect to Hk for k ∈ {0, 1, . . . , n}.
iii) A third question is about the preservation of the SBV-regularizing effect in the case of
a Hamiltonian which is only convex. In this case the property of semiconcavity of the
solution is lost and its Hessian is no more a measure. However one can try to prove a
kind of SBV regularity for the Radon measure divHp (Dx u(t, ·)). A positive answer will
generalize Theorem 0.2 because for semiconcave functions the Cantor part of Dx2 u(t, ·) is
controlled by the Cantor part of the spatial Laplacian ∆u(t, ·).
iv) One can look for applications of Theorem 0.2. In the one-dimensional case an easy application follows for Convection Theory and systems of sticky particles. One can wonder if
that theorem applies also in the multi-dimensional case.
10
Introduction
Let us consider in more details all these cases.
i) In Chapter 2 we consider the general Hamilton-Jacobi equation (1), introduced at the
beginning of the section, and we require the following assumptions on H:
(H1) H ∈ C 3 ([0, T ]×Rn ×Rn ) with bounded second derivatives and there exist positive constants
a, b, c such that
i) H(t, x, p) ≥ −c,
ii) H(t, x, 0) ≤ c,
iii) |Hpx (t, x, p)| ≤ a + b|p|,
(H2) there exists cH > 0 such that
c−1
H Idn (p) ≤ Hpp (t, x, p) ≤ cH Idn (p)
for any t ∈ R, x ∈ Rn .
As already seen these assumptions are necessary to relate our equation to a well defined problem
in Calculus of Variations. The idea is to reproduce the strategy seen for the case of a Hamiltonian
H = H(p). The main difference from that case is that, due to the dependence of the Hamiltonian
on (t, x), characteristics are curves in C 2 but in general they are not straight lines. This is
practically the main difficulty to overcome since the strategy used in [9] heavily takes advantage
on the simple form of characteristics, while in this case it is not so easy to find an explicit
form for them. However, considering sufficiently small intervals of time, one can approximate
characteristics with straight lines. Generalized backward characteristics ξ are solutions, together
with their dual arc p, of the system
{
˙
ξ(s)
= Hp (s, ξ(s), p(s))
(4)
ṗ(s) = −Hx (s, ξ(s), p(s))
{
with final conditions
ξ(t) = x
p(t) = p,
where p ∈ Dx+ u(t, x). We will show that one can establish a one to one correspondence between
generalized backward characteristics and maximizers of the backward solution
{
}
∫ t
−
2
n
˙
ut,0 (τ, y) := max u(t, ξ(t)) −
L(s, ξ(s), ξ(s))ds
ξ(τ ) = y, ξ ∈ [C ([τ, t])] .
τ
Thus the map Xt,τ (x), now defined as
Xt,τ (x) := {ξ(τ )| ξ(·) is a solution of (4), with ξ(t) = x, p(t) = p ∈ Dx+ u(t, x)},
over the small interval of time [τ, τ + ε], is injective. Handling with care the difference between
characteristics and straight lines in a small interval of time, it is possible to recover the nocrossing property and the two area estimates up to an error of the order of ε. Therefore the
usual strategy can be easily adapted.
Thus we are able to prove the following theorem.
11
Introduction
Theorem 0.3. Let u be a viscosity solution of (1), assume H1, H2. Then the set of times
S := {t | Dx u(t, ·) ̸∈ [SBVloc (Ωt )]n }
is at most countable. In particular Dx u ∈ [SBVloc (Ω)]n , ∂t u ∈ SBVloc (Ω).
The results presented in Chapter 2 are contained in Bianchini and Tonon [13].
ii) In Chapter 3 we present one of the consequences of Theorem 0.2. In the case of a
uniformly convex Hamiltonian, the Jacobian J(t, ·) := Hn (Dx+ u(t, ·)), defined on Ωt , has a
particular structure. Out of a countable number of t’s, the measure J(t, ·) has only an absolute
continuous part with respect to Hn and a part which is concentrated on a Hn−1 -rectifiable set
and is absolute continuous with respect to Hn−1 . This is a direct consequence of the fact that
|Dx2 u(t, ·)| can have Cantor part for a countable number of t’s only. This fact suggests that the
Jacobian measure has a structure which admits only integer parts. That is, out of a countable
number of t’s, J(t, ·) can have only parts which are concentrated on a Hk -rectifiable set and are
absolute continuous with respect to Hk for k ∈ {0, 1, . . . , n}. However, a counterexample in R2
shows that this is not true in general. It is possible to find a viscosity solution whose Jacobian
has a positive part between H1 and H0 .
iii) In Chapter 4 we consider the Hamilton-Jacobi equation (3) with a convex Hamiltonian
H. When H is smooth and only convex, the Lagrangian L is strictly convex but no more
regular. Therefore u(t, ·) is no more semiconcave and Dx u(t, ·) looses its BV regularity. The only
regularity which is true in general for the viscosity solution u is local Lipschitzianity. Thus there
is no hope to prove the SBV-regularizing effect for Dx u(t, ·) apart from some particular cases.
However, a kind of SBV regularity can be proven for the vector field d(t, x) := Hp (Dx u(t, x)).
This vector field is defined only on the set of points (t, x) where u(t, x) is differentiable in x but
can be extended to the all Ω using the optimal rays of the forward solution. Once the vector
field is extended, its divergence divd(t, ·) is shown to be a locally finite Radon measure. It is
therefore reasonable to see if this measure admits a Cantor part for all t. When the vector field
d(t, ·) is BV and suitable hypotheses on the Lagrangian L are made, the measure divd(t, ·) has
Cantor part only for a countable number of t’s in [0, T ].
The strategy to obtain our result was suggested by an extension of Theorem 0.1, done by
Robyr in [35], for the scalar conservation law
∂t U (t, x) + Dx (H(t, x, U (t, x))) + g(t, x, U (t, x)) = 0
in Ω ⊂ R+ × R.
(5)
Theorem 0.4 (Robyr). Let H ∈ C 2 (R+ × R × R) be a flux function, such that
{pi ∈ R| Hpp (t, x, pi ) = 0}
is at most countable for any fixed (t, x). Let g ∈ C 1 (R+ × R × R) be a source term and let
U ∈ BV (Ω) be an entropy solution of the balance law (5).
Then there exists a set S ⊂ R+ at most countable such that ∀t ∈ R+ \ S the following holds:
U (t, ·) ∈ SBVloc (Ωt ).
In particular U (t, x) belongs to SBVloc (Ω).
12
Introduction
First we prove that, in the one-dimensional case, the BV regularity of d(t, x) = Hp (Dx u(t, x))
follows automatically in the case of a convex smooth H and there is no need to add hypotheses to
prove its SBV regularity out of a countable number of t’s. This fact however does not necessarily
implies that the same apply to Dx u(t, x) = U (t, x), its BV regularity remains not true in general
even in the one-dimensional case.
Moreover the proof of Theorem 0.4 suggested to us an idea to cope with the fact that H is
only convex in the multi-dimensional case. As already said, when H is smooth and convex L is
not C 2 in general. However, since we are looking at the Cantor part of divd(t, ·), we can reduce
to the set U := {(t, x)| u(t, x) is differentiable in x}. Indeed the set Ωt \ Ut has null Hn -measure
for every t and (divd(t, ·))c (Ωt \ Ut ) = 0. Moreover we can consider separately the set of points
(t, x) ∈ U where L(d(t, x)) is C 2 and the set of points where L(d(t, x)) is not twice differentiable.
In the first set we reduce locally to the uniformly convex case. Thus we can apply Theorem
0.2 to obtain the SBV-regularity. In the second, we need to add some hypotheses to handle
the problem. Since we are able to prove the regularity of divd(t, ·) in the one-dimensional case
the idea is to reduce step by step to dimension one. A way to do this is to require that the
vector field d(t, ·) is BV and that the Lagrangian is such that the set of points where its Hessian
is not defined is contained in a finite number of hyperplanes. We can then study our problem
restricted to these hyperplanes reducing the dimension to n − 1. Repeating the procedure, we
need to ask the following hypotheses.
Let H be C 2 (Rn ), convex and such that lim|p|→∞ H(p)
|p| = +∞.
(HYP(0)) Suppose the vector field d(t, ·) belongs to [BV (Ωt )]n for every t ∈ [0, T ].
Define Vπn as
Vπn := {v ∈ Rn | L(·) is not twice differentiable in v},
and
Σπn := {(t, x) ∈ U | d(t, x) ∈ Vπn }
and Σcπn := U \ Σπn .
(HYP(n)) We suppose Vπn to be contained in a finite union of hyperplanes Ππn .
For j = n, . . . , 3 for every (j − 1)-dimensional plane πj−1 in Ππj , let Lπj−1 : Rj−1 → R be
the (j − 1)-dimensional restriction of L to πj−1 and
Vπj−1 := {v ∈ Rj−1 | Lπj−1 (·) is not twice differentiable in v}.
Define
Σπj−1 := {(t, x) ∈ Σπj | d(t, x) ∈ Vπj } and
Σcπj−1 := Σπj \ Σπj−1 .
(HYP(j-1)) We suppose Vπj−1 is contained in a finite union of (j − 2)-dimensional planes Ππj−1 ,
for every πj−1 ∈ Ππj .
Theorem 0.5. Under the above assumptions (HYP(0)),(HYP(n)),...,(HYP(2)), the Radon
measure divd(t, ·) has Cantor part on Ωt only for a countable number of t’s in [0, T ].
The question on the SBV regularity of d(t, ·) without any additional hypothesis to the convexity of H is still open.
The results presented in Chapter 4 are contained in Bianchini and Tonon [12].
13
Introduction
iv) In Chapter 5 we present an application of Theorem 0.1. In the one-dimensional case
the Generalized Hydrostatic Boussinnesq equations of Convection Theory and sticky particles
systems can be both described at a discrete level by a finite collection of particles that get
stuck together right after they collide. At a continuous level, instead, they can be related to a
scalar conservation law of type (2 with non decreasing initial datum and bounded Lipschitz flux
function. When U0 (x) := x is chosen as initial datum we can reduce to an equivalent HamiltonJacobi equation where the flux function is 12 |x|2 and the initial datum is H. Therefore the result
of Ambrosio and De Lellis can be used to prove the SBV regularity of the entropy solution of
the scalar conservation law with non decreasing initial datum U0 (x) := x and bounded Lipschitz
flux function.
Considering the multi-dimensional case it is reasonable to ask whether Hamilton-Jacobi
equations are again related to the multidimensional version of the Generalized Hydrostatic
Boussinnesq equations of Convection Theory and sticky particles systems. Theorem 0.2 could be
applied even to this case if the answer was affirmative. However, in the multi-dimensional case,
Hamilton-Jacobi equations are no more able to describe Convection Theory and sticky particles
systems. Indeed, viscosity solutions of the multi-dimensional Hamilton-Jacobi equations can
behave in a way which is not allowed in Convection Theory and by sticky particles systems.
For viscosity solutions, particles that collide eventually separate while in Convection Theory
and in sticky particles systems particles get stuck together after a collision. We present a
counterexample which shows precisely this behavior. A first counterexample was found by
Vasseur in his PhD Thesis [38] but it was never published.
The results presented in Chapter 5 are contained in Tonon [37].
0.2
Decomposition for BV functions
One of the necessary and sufficient properties, which characterizes real valued BV functions of
one variable, is the well-known Jordan decomposition: it states that a function f : R → R is of
bounded variation if and only if it can be written as the difference of two monotone increasing
functions.
In Chapter 6, we present a generalization of this property to real valued BV functions of
many variables.
To this aim, we define a new concept of monotonicity for a real valued function of many
variables. Many different definitions of monotone function already exist in literature.
One can in fact preserve the monotonicity of the product ⟨f (x) − f (y), x − y⟩ ≥ 0, defining
that f : Rn → Rn is monotone if
⟨f (x) − f (y), x − y⟩ ≥ 0,
where ⟨·, ·⟩ is the scalar product in Rn .
Another possibility is to preserve the maximum principle: the supremum (infimum) of f in
every set is assumed at the boundary. Taken Ω ⊂ Rn , a Lebesgue monotone function is defined as
a continuous function f : Ω → R, which satisfies the maximum and minimum principles in every
subdomain. Manfredi, in [33], and Hajlasz and Malý, in [31], give a weaker formulation. Here,
a weakly monotone function is defined as a function f : Ω → R in the Sobolev space W 1,p (Ω),
14
0.2 Decomposition for BV functions
which satisfies the weak maximum and the weak minimum principles in every subdomain. A
1,p
natural generalization is given in the case f is in the Sobolev space Wloc
(Ω).
In our case we choose to define monotone a function whose sub-level and super-level sets are
indecomposable and of finite perimeter for H1 -a.e. t ∈ R.
This notion of monotonicity is no more a sufficient condition for a function to be of bounded
variation, as it was in the Jordan decomposition. However, it allows a decomposition of BV
functions in a countable sum of monotone functions.
Indeed, in the case of BV functions, sub-level and super-level sets are of finite perimeter
for H1 -a.e. t ∈ R. Moreover, sets with finite perimeter can be decomposed in the countable
union of indecomposable sets, up to Hn -negligible sets, see [3]. Therefore, playing with the
indecomposable components of sub-level and super-level sets, one can decompose a BV function
in the sum of monotone BV functions. In general more than one of such decompositions is
possible.
The strategy above is the idea which lies in of the proof of the following theorem.
Theorem 0.6. Let f : Rn → R be a BV (Rn ) function. Then there exists a finite or countable
family of monotone BV (Rn ) functions {fi }i∈I , such that
f=
∑
fi
and
|Df | =
i∈I
∑
|Dfi |.
i∈I
This result extends a theorem of Alberti, Bianchini and Crippa presented in Section 6.1
which proves a decomposition property for real valued Lipschitz functions of many variables.
Theorem 0.7 (Alberti, Bianchini and Crippa). Let f be a function in Lipc (Rn ) with compact
support. Then there exists a countable family {fi }i∈N of functions in Lipc (Rn ) such that f =
∑
i fi and each fi is monotone. Moreover there is a pairwise disjoint partition {Ωi }i∈N of Borel
sets of Rn such that ∇fi is concentrated on Ωi .
In the case of a Lipschitz function monotonicity is given by the connectedness of level sets.
Moreover the decomposition preserves the mutual singularity of every ∇fi . This is no more
true in the BV case. It can be found an example where the monotone functions given by the
decomposition have distributional derivatives which in general are not mutually singular.
Theorem 0.6 is in a way optimal. We show with a counterexample that there is no hope for a
further generalization of this decomposition to vector valued BV functions, apart from the case
of a function f : R → Rm where the analysis is straightforward. Indeed, a Lipschitz function
from R2 to R2 can be decomposed in a sum of monotone functions only if some of its level sets
are of positive H1 -measure. This is an additional property, which is clearly not shared by all
the Lipschitz functions.
The results presented in Chapter 6 are contained in Bianchini and Tonon [11].
15
CONTENTS
0.3
Notations
Hn
R+
[L1 (Rn )]m
L1loc (Rn )
[Lipc (Rn )]m
[BV (Rn )]m
∇f
Df
|Df |
Dx u(t, x)
∂t u(t, x)
divd(t, ·)
Hp (t, x, p)
Hpp (t, x, p)
Hpx (t, x, p)
⟨·, ·⟩
O(t)
Et
Ex
P (E)
|x|
∥f ∥V
E̊ M
E
χE
(mod Hn )
δx
∂E
d(·, ·)
dH (A, B)
n-dimensional Hausdorff measure
set of all non negative real number
Lebesgue space of functions from Rn to Rm
space of functions from Rn to R which are locally L1 (Rn )
space of c-Lipschitz functions from Rn to Rm
space of bounded variation functions from Rn to Rm
gradient of the Lipschitz function f
distributional derivative of the BV function f
total variation of the function f
spatial distributional derivative of the locally Lipschitz function u(t, x)
time distributional derivative of the locally Lipschitz function u(t, x)
spatial divergence of the vector field d(t, ·)
derivative with respect to p
second derivative with respect to p
derivative with respect to p and with respect to x
scalar product in Rn
big O notation
{x ∈ Rn | (t, x) ∈ E} for E ⊂ R+ × Rn
{y ∈ Rn | (x, y) ∈ E} for E ⊂ Rn+k , x ∈ Rk
perimeter of the set E
norm of the vector x ∈ Rn
norm of a function in the space V
essential interior of the set E
closure of the set E
characteristic function of the set E
up to HN -negligible sets
Dirac measure
topological boundary of a set E
Euclidean distance
Hausdorff distance between the sets A and B
16
Chapter 1
Preliminaries
We list here some preliminary results which will be necessary in the following chapters. References with more detailed descriptions of the arguments treated can be found therein.
1.1
Rectifiable sets
We briefly introduce the concept of rectifiable sets, for a more comprehensive reference see [5].
Definition 1.1. An Hk -measurable set E ⊂ Rn is Hk -rectifiable if there exist countably many
Lipschitz functions fi : Rk → Rn such that
(
)
∞
∪
Hk E \
fi (Rk ) = 0
i=0
and Hk (E) < ∞.
1.2
Hausdorff distance and Hausdorff convergence of sets
Let d(·, ·) be the Euclidean distance in Rn .
Definition 1.2. Given a set A ⊂ Rn and a point x ∈ Rn their distance is defined as
d(x, A) := inf d(x, y).
y∈A
Rn
Definition 1.3. Given two sets A, B ⊂
{
then their Hausdorff distance is defined as
}
dH (A, B) := max sup inf d(x, y), sup inf d(x, y) ,
x∈A y∈B
y∈B x∈A
or equivalently
dH (A, B) = inf{ε > 0| A ⊂ Bε and B ⊂ Aε },
where
Eε := ∪x∈E {z ∈ Rn | d(x, z) < ε}
for any set E ⊂ Rn .
17
Preliminaries
Definition 1.4. A sequence of sets E ε ⊂ Rn converges in the Hausdorff sense to E ⊂ Rn if
lim dH (E ε , E) = 0.
ε→0
1.3
Generalized differentials
We recall the definition of generalized differential, see Cannarsa and Sinestrari [20] and Cannarsa
and Soner [21].
In this section Ω will be an open subset of Rn .
Definition 1.5. Let u : Ω → R, for any x ∈ Ω the sets
}
{
u(y) − u(x) − ⟨p, y − x⟩
−
n
≥0 ,
D u(x) = p ∈ R | lim inf
y→x
|y − x|
{
}
u(y) − u(x) − ⟨p, y − x⟩
n
D u(x) = p ∈ R | lim sup
≤0 ,
|y − x|
y→x
+
are called, respectively, the subdifferential and superdifferential of u at x.
D+ u and D− u are closed and convex sets, sometimes they can be empty.
Definition 1.6. Let u : Ω → R be locally Lipschitz. A vector p ∈ Rn is called a reachable
gradient of u at x ∈ Ω if there exists a sequence {xk } ⊂ Ω \ {x} such that u is differentiable at
xk for each k ∈ N, and
lim xk = x,
lim Du(xk ) = p.
k→∞
k→∞
The set of all reachable gradients of u at x is denoted by D∗ u(x).
1.4
Decomposition of a Radon measure
Given an [L∞ (Rn )]n vector field d(x) such that divd(x) =: µ(x) is a Radon measure on Rn , we
can decompose µ into three mutually singular measures:
µ = µa + µc + µj .
µa is the absolutely continuous part with respect to the Lebesgue measure. µj is the singular
part of the measure which is concentrated on a Hn−1 -rectifiable set. µc , the Cantor part, is the
remaining part.
1.5
BV and SBV functions
A detailed description of the spaces BV and SBV can be found in Ambrosio, Fusco and Pallara
[5], Chapters 3 and 4.
18
1.5 BV and SBV functions
Definition 1.7. A function u : Rn → R, which belongs to L1 (Rn ), is said to be of bounded
variation if its distributional derivative is representable as an Rn -valued measure Du with finite
total variation, i.e.
∫
n ∫
∑
udivϕdx = −
ϕi dDi u ∀ϕ ∈ [Cc1 (Rn )]n .
Rn
i=1
Rn
The total variation |Du| of a BV function is defined as the total variation of the vector measure
Du. A function u : Rn → Rk is said to be of bounded variation if every components uj : Rn → R
is of bounded variation for j = 1, . . . , k.
Given u ∈ BV (Rn ), it is possible to decompose the distributional derivative of u into three
mutually singular measures:
Du = Da u + Dc u + Dj u.
Da u is the absolutely continuous part with respect to the Lebesgue measure. Dj u is the part of
the measure which is concentrated on the rectifiable (n−1)-dimensional set J, where the function
u has “jump” discontinuities, thus for this reason it is called jump part. Dc u, the Cantor part,
is the singular part which satisfies Dc u(E) = 0 for every Borel set E with Hn−1 (E) < ∞. If this
part vanishes, i.e. Dc u = 0, we say that u ∈ SBV (Rn ). When u ∈ [BV (Rn )]k the distributional
derivative Du is a matrix of Radon measure and the decomposition can be applied to every
component of the matrix.
We recall here some properties of BV functions which will be useful later on.
Definition 1.8. Let u in [L1loc (Rn )]k , we say that u has an approximate limit at x ∈ Rn if there
exists z ∈ Rk such that
lim
|u(y) − z|dy = 0.
ρ→0 Bρ (x)
The set Su of points where this property does not hold is called the approximate discontinuity
set. For any x ∈ Rn \ Su the vector z is called approximate limit of u at x and is denoted by
ũ(x).
Proposition 1.9. Let u and v belong to [BV (Rn )]k . Let
L := {x ∈ Rn \ (Su ∪ Sv )| ũ(x) = ṽ(x)}.
Then Du and Dv are equal when restricted to L.
Proof. See Remark 3.93 in [5].
Proposition 1.10. Let u belongs to [BV (Rn )]k . Then Dc u vanishes on sets which are σ-finite
with respect to Hn−1 and on sets of the form ũ−1 (E) with E ⊂ Rk and H1 (E) = 0.
Proof. See Proposition 3.92 in [5].
Proposition 1.11. Let u belongs to [BV (Rn )]k . For j = 1, . . . , n − 1 define the (n − j)dimensional restriction ux1 ,...,xj (·) : Rn−j → Rk as ux1 ,...,xj (x̂) = u(x1 , . . . , xj , x̂) for fixed
x1 , . . . , xj ∈ Rj . Then ux1 ,...,xj (·) is [BV (Rn−j )]k for Hj -a.e. x1 , . . . , xj in Rj .
Proof. This is a well known result. The proof in the case j = n − 1 can be found in [5] Section
3.11, in the other cases is similar.
19
Preliminaries
1.6
Semiconcave functions
For a complete introduction to the theory of semiconcave functions we refer to Cannarsa and
Sinestrari [20], Chapter 2 and 3 and Lions [32]. For our purpose we define semiconcave functions
with a linear modulus of semiconcavity. In general this class is considered only as a particular
subspace of the class of semiconcave functions with general semiconcavity modulus. The proofs
of the following statements can be found in the mentioned references.
In this section Ω will be an open subset of Rn .
Definition 1.12. We say that a function u : Ω → R is semiconcave and we denote with SC(Ω)
the space of functions with such a property, if for a C > 0 and for any x, z ∈ Ω such that the
segment [x − z, x + z] is contained in Ω
u(x + z) + u(x − z) − 2u(x) ≤ C|z|2 .
Proposition 1.13. Let u : Ω → R belongs to SC(Ω) with semiconcavity constant C ≥ 0. Then
the function
C
ũ : x 7→ u(x) − |x|2
2
is concave, i.e. for any x, y in Ω such that the whole segment [x, y] is contained in Ω, λ ∈ [0, 1]
ũ(λx + (1 − λ)y) ≥ λũ(x) + (1 − λ)ũ(y).
Theorem 1.14. Let u : Ω → R belongs to SC(Ω). Then the following properties hold.
i) (Alexandroff ’s Theorem) u is twice differentiable Hn -a.e.; that is, for Hn -a.e. x0 ∈ Ω,
there exist a vector p ∈ Rn and a symmetric matrix M such that
lim
x→x0
u(x) − u(x0 ) − ⟨p, x − x0 ⟩ + ⟨M (x − x0 ), x − x0 ⟩
= 0.
|x − x0 |2
ii) The gradient of u, defined Hn -a.e. in Ω, belongs to the class BVloc (Ω, Rn ).
iii) Let x ∈ Ω then
D+ u(x) = coD∗ u(x),
where coA := min{B | B ⊃ A, B convex} is the convex hull of A. Thus D+ u is non empty
at each point. Moreover D+ u is upper semicontinuous.
iv) The function T (x) := −D+ ũ(x) is a maximal monotone function, i.e.
⟨y1 − y2 , x1 − x2 ⟩ ≥ 0
∀xi ∈ Ω yi ∈ T (xi ) i = 1, 2;
and it is maximal in following sense
V ⊃ T, V monotone =⇒ V = T.
20
1.7 Viscosity solutions
As stated in the above theorem at point ii), when u is semiconcave Du is a BV map, hence
the distributional Hessian D2 u is a symmetric matrix of Radon measures and can be split into
the three mutually singular parts Da2 u, Dj2 u, Dc2 u. Moreover the following proposition holds.
Proposition 1.15. Let u be a semiconcave function. If D denotes the set of points where D+ u
is not single-valued, then |Dc2 u|(D) = 0.
Proof. Indeed, the set of points where D+ u is not single-valued, i.e. the set of singular points,
is a Hn−1 -rectifiable set.
Definition 1.16. We say that a function v : Ω → R is semiconvex if u := −v is semiconcave.
1.7
Viscosity solutions
A concept of generalized solutions to the equations
∂t u + H(t, x, Dx u) = 0
in Ω ⊂ [0, T ] × Rn ,
(1.1)
and
in Ω ⊂ Rn ,
H(x, Du) = 0
(1.2)
was found to be necessary since classical solutions break down and solutions which satisfy (1.1)
almost everywhere are not unique. Crandall and Lions introduced in [23] the notion of viscosity
solution to solve both these problems, see also Crandall, Evans and Lions [22]. A viscosity
solution needs not be differentiable anywhere, the only regularity required in the definition is
uniform continuity. This concept ensures existence, stability and uniqueness of solutions for a
wider class of equations.
Definition 1.17. A bounded uniformly continuous function u : Ω → R is called a viscosity
solution of (1.1) (resp. (1.2)) provided that
i) u is a viscosity subsolution of (1.1) (resp. (1.2)): for each v ∈ C ∞ (Ω) such that u − v has
a maximum at (t0 , x0 ) ∈ Ω (resp. x0 ∈ Ω),
∂t v(t0 , x0 ) + H(t0 , x0 , Dx v(t0 , x0 )) ≤ 0
(resp. H(x0 , Dv(x0 )) ≤ 0);
ii) u is a viscosity supersolution of (1.1) (resp. (1.2)): for each v ∈ C ∞ (Ω) such that u − v
has a minimum at (t0 , x0 ) ∈ Ω (resp. x0 ∈ Ω),
∂t v(t0 , x0 ) + H(t0 , x0 , Dx v(t0 , x0 )) ≥ 0
1.8
(resp. H(x0 , Dv(x0 )) ≥ 0).
Properties of viscosity solutions when H = H(t, x, p)
We will consider here only viscosity solutions of equation (1.1), similar results apply also to
viscosity solutions of the Hamilton-Jacobi equation (1.2).
Let us introduce a locality property.
21
Preliminaries
Proposition 1.18. Let u be a viscosity solution of (1.1) in Ω, when the Hamiltonian H is
convex in the last variable. Then u is locally Lipschitz. Moreover for any (t0 , x0 ) ∈ Ω, there
exists a neighborhood U of (t0 , x0 ), a positive number δ and a Lipschitz function v0 on Rn such
that
(Loc) u coincides on U with the viscosity solution of
{
∂t v + H(t, x, Dx v) = 0
in [t0 − δ, ∞) × Rn
v(t0 − δ, x) = v0 (x).
Proof. The proof of Proposition 3.5, given in [9], still applies in our case. We only loose the
property that minimizers of the representation formula for the viscosity solution (see later on
1.3) are straight lines which was unnecessary for the argument. Even the uniform convexity of
H in the last variable was not necessary in the proof.
Motivated by the above proposition, it is enough to consider the Cauchy problem
{
∂t u + H(t, x, Dx u) = 0 in Ω ⊂ [0, T ] × Rn ,
u(0, x) = u0 (x) for all x ∈ Ω0 ,
where u0 (x) is a bounded Lipschitz function on Ω0 := {x ∈ Rn | (0, x) ∈ Ω}.
The proofs of the following statements can be found in Cannarsa and Sinestrari [20], Chapter
6. See also Fleming [28], Fleming and Rishel [29], Fleming and Soner [30] and Lions [32].
The convexity of the Hamiltonian in the p-variable relates Hamilton-Jacobi equations to a
variational problem.
Let us require the following assumptions on H.
(H1) H ∈ C 3 ([0, T ]×Rn ×Rn ) with bounded second derivatives and there exist positive constants
a, b, c such that
i) H(t, x, p) ≥ −c,
ii) H(t, x, 0) ≤ c,
iii) |Hpx (t, x, p)| ≤ a + b|p|,
(H2) H is uniformly convex in the last variable, to be more precise there exists cH > 0 such
that
c−1
H Idn (p) ≤ Hpp (t, x, p) ≤ cH Idn (p)
for any t ∈ R, x ∈ Rn .
Let L be the Lagrangian of our system, i.e. the Legendre transform of the Hamiltonian H
with respect to the last variable, for any t, x fixed
L(t, x, v) = sup{⟨v, p⟩ − H(t, x, p)}.
p
The Legendre transform inherits the properties of H, in particular L is C 3 ([0, T ] × Rn × Rn )
and uniformly convex in the last variable.
In addition to the uniform convexity and C 3 regularity of L, the hypotheses on H, (H1) and
(H2), ensure the existence of positive constants a, b, c such that
22
1.8 Properties of viscosity solutions when H = H(t, x, p)
i) L(t, x, v) ≥ −c,
ii) Lx (t, x, 0) ≤ c,
iii) |Lvx (t, x, v)| ≤ a + b|v|.
Define the value function u(t, x) associated the bounded Lipschitz function u0 (x), for (t, x) ∈
Ω
{
∫
u(t, x) := min u0 (ξ(0)) +
0
t
}
2
n
˙
L(s, ξ(s), ξ(s))ds ξ(t) = x, ξ ∈ [C ([0, t])] .
(1.3)
Less regularity can be asked to ξ, but it is unnecessary since any minimizing curve exists
and is smooth, due to the regularity of L, see [20].
Theorem 1.19. Taken a minimizing curve ξ in (1.3), for the point (t, x), such that ξ(s) ∈ Ωs
for all s ∈ [0, t], the following holds.
˙
i) The map s 7→ Lv (s, ξ(s), ξ(s))
is absolutely continuous.
ii) ξ is a classical solution to the Euler-Lagrange equation
d
˙
˙
Lv (s, ξ(s), ξ(s))
= Lx (s, ξ(s), ξ(s)),
ds
and to the Du Bois-Reymond equation
d
˙
˙
˙
˙
[L(s, ξ(s), ξ(s))
− ⟨ξ(s),
Lv (s, ξ(s), ξ(s))⟩]
= Lt (s, ξ(s), ξ(s)),
ds
˙
˙
˙
for all s ∈ [0, t], where Lt (s, ξ(s), ξ(s)),
Lx (s, ξ(s), ξ(s)),
Lv (s, ξ(s), ξ(s))
are the derivatives
of L with respect to t, x, v respectively.
iii) For any r > 0 there exists K(r) > 0 such that, if (t, x) ∈ [0, r] × Br (0), then
˙
sup |ξ(s)|
≤ K(r).
s∈[0,t]
iv) There exists a dual arc or co-state
˙
p(s) := Lv (s, ξ(s), ξ(s))
s ∈ [0, t],
(1.4)
such that ξ, p solve the following system
{
˙
ξ(s)
= Hp (s, ξ(s), p(s))
ṗ(s) = −Hx (s, ξ(s), p(s)).
v) (s, ξ(s)) is regular, i.e. for any 0 < s < t, ξ is the unique minimizer for u(s, ξ(s)), and
u(s, ·) is differentiable at ξ(s).
23
Preliminaries
vi) Let p be the dual arc associated to ξ as in (1.4) then we have
p(t) ∈ Dx+ u(t, x),
p(s) = Dx u(s, ξ(s)),
s ∈ (0, t).
Theorem 1.20. The value function u defined in (1.3) is a viscosity solution of (1.1) with
bounded Lipschitz initial datum
u(0, x) = u0 (x).
Definition 1.21. A point (t, x) ∈ Ωt is called regular if there exists a unique minimizer for
u(t, x). All the other points are called irregular.
A point (t, x) ∈ Ωt is called conjugate if z ∈ Rn exists such that ξ(t, z) = x, ξ(·, z) is a
minimizer for u(t, x) and
det ξz (t, z) = 0.
We present below some properties of the unique viscosity solution to the Hamilton-Jacobi
equation (1.1), which follow from the representation formula we have just seen. These properties
are taken from [20].
Theorem 1.22 (Dynamic Programming Principle). Fix (t, x), then for all t′ ∈ [0, t]
}
{
∫ t
2
′
n
′
′
˙
L(s, ξ(s), ξ(s))ds
u(t, x) := min u(t , ξ(t )) +
ξ(t) = x, ξ ∈ [C ([t , t])] .
t′
(1.5)
Moreover if ξ is a minimizer in (1.3) it is a minimizer also for (1.5) for any t′ ∈ [0, t].
Theorem 1.23. Suppose (H1), (H2) hold and u0 belongs to Cb (Rn ). Then for any t in (0, T ],
u(t, ·) is locally semiconcave with semiconcavity constant C(t) = Ct . Thus for any fixed τ > 0
there exists a constant C = C(τ ) such that u(t, ·) is semiconcave with constant less than C for
any t ≥ τ .
Moreover u is also locally semiconcave in both the variables (t, x) in (0, T ] × Rn .
1.8.1
Minimizers and Generalized Backward Characteristics
We introduce the definition of generalized backward characteristics.
Definition 1.24. Given x ∈ Ωt for t fixed in [0, T ], we call generalized backward characteristic,
associated to u starting from x, the curve s 7→ (s, ξ(s)), where ξ(·) and its dual arc p(·) solve
the system
{
˙
ξ(s)
= Hp (s, ξ(s), p(s))
(1.6)
ṗ(s) = −Hx (s, ξ(s), p(s))
with final conditions
{
ξ(t) = x
p(t) = p,
where p ∈ Dx+ u(t, x).
If Dx+ u(t, x) is single-valued then we call ξ a classical backward characteristic.
24
(1.7)
1.8 Properties of viscosity solutions when H = H(t, x, p)
We state here some properties of minimizers which strictly relate them with classical and
generalized characteristics, see [20].
Theorem 1.25. For any (t, x) ∈ Ω the map that associates with any (pt , px ) ∈ D∗ u(t, x) the
curve ξ obtained by solving the system (1.6) with the final conditions
{
ξ(t) = x
p(t) = px
provides a one-to-one correspondence between D∗ u(t, x) and the set of minimizers of u(t, x).
Thus we can state the following theorem which follows from Theorem 1.19-(iv), Theorem
1.25 and Definition 1.24.
Theorem 1.26. Let (t, x) in Ω be given, and let ξ be a C 2 curve such that ξ(s) ∈ Ωs for all
0 ≤ s ≤ t.
Then ξ is a minimizer if and only if ξ and its dual arc p are solutions of the system (1.6)
for any s ∈ [0, t] with final conditions (1.7), where (−H(t, x, p), p) belongs to D∗ u(t, x).
A minimizer ξ is a generalized backward characteristic. In particular ξ is a classical backward
characteristic if and only if ξ is the unique minimizer for u(t, x). The set of minimizers for u(t, x)
is a proper subset of the set of generalized backward characteristics emanated from (t, x).
Remark 1.27. Note that, the solutions ξ of the system (1.6) are in general curves and not
straight lines, as in the case H = H(p).
Remark 1.28. No-crossing property of minimizers. Fix a time t and consider a minimizing
curve ξ such that ξ(t) = x ∈ Ωt . For 0 < s < t the curve ξ is the unique minimizer for
u(s, ξ(s)), this ensures that any other minimizer cannot intersect ξ for any 0 < s < t (otherwise
uniqueness would be lost, see point (v) of Theorem 1.19). As a consequence generalized backward
characteristics which are also minimizers, i.e. solution of (1.6), (1.7), where (−H(t, x, p), p)
belongs to D∗ u(t, x), cannot intersect except than in 0 or t. Nothing can be said at this level
for generalized backward characteristics solution to (1.6) with
p(t) = p ∈ Dx+ u(t, x) \ Dx∗ u(t, x),
ξ(t) = x
which are not minimizers. In general they can cross.
1.8.2
Backward solutions
The introduction of a backward solution, as in Barron, Cannarsa, Jensen and Sinestrari [7],
will allow us to see in Section 2.1 that, at least for a small interval of time, all the generalized
backward characteristics share the no-crossing property.
Fix t in (0, T ] and define for 0 ≤ τ < t, y ∈ Ωτ the function
{
}
∫ t
−
2
n
˙
ut,0 (τ, y) := max u(t, ξ(t)) −
L(s, ξ(s), ξ(s))ds ξ(τ ) = y, ξ ∈ [C ([τ, t])] .
(1.8)
τ
Note that the function v(τ, y) :=
u−
t,0 (t
− τ, y) is a viscosity solution of
∂τ v − H(t − τ, y, Dy v) = 0
in Ω ⊂ [0, T ] × Rn
−
with initial datum v(0, y) = u−
t,0 (t, y) = u(t, y), for this reason ut,0 is called backward solution.
25
Preliminaries
Proposition 1.29. In general
u−
t,0 (τ, y) ≤ u(τ, y)
and the equality holds if and only if the maximizer ξ in (1.8), defined for τ ≤ s ≤ t, is part of a
minimizing curve for u(t, ξ(t)).
Proof. Let ξ be a C 2 -curve which is a maximizer for u−
t,0 (τ, y), i.e.
u−
t,0 (τ, y)
∫
= u(t, ξ(t)) −
t
˙
L(s, ξ(s), ξ(s))ds.
τ
Thanks to the Dynamic Programming Principle,
∫ t
˙
u(t, ξ(t)) ≤ u(τ, y) +
L(s, ξ(s), ξ(s))ds.
τ
Hence,
u−
t,0 (τ, y) ≤ u(τ, y)
and the equality holds if and only if ξ is also a minimizer for u(t, ξ(t)), thus Dx+ u(s, ξ(s)) is
single-valued for any τ ≤ s < t.
Note that a curve ξ which is a minimizer for u(t, x) is also a maximizer for u−
t,0 (τ, ξ(τ )) =
u(τ, ξ(τ )) for any 0 ≤ τ < t.
With suitable modifications Theorems 1.19, 1.20, 1.22 and 1.23 still hold for u−
t,0 (τ, y) and
−
C
.
its maximizers, in particular ut,0 is semiconvex (rather than semiconcave) with constant t−τ
Without adding any other assumption, the no-crossing property holds also for maximizers.
1.9
Properties of viscosity solutions when H = H(p)
We present here the case in which H = H(p) is smooth and convex. Some of the results are just
a particular case of the results obtained in the case H = H(t, x, p).
As already noticed in the case of H = (t, x, p) it is enough to consider the Cauchy problem
{
∂t u + H(Dx u) = 0 in Ω ⊂ [0, T ] × Rn ,
(1.9)
u(0, x) = u0 (x) for all x ∈ Ω0 ,
where u0 (x) is a bounded Lipschitz function on Ω0 .
The proofs of the following statements can be found in Evans [26], Section 3.3 and Chapter
10. See also Cannarsa and Sinestrari [20], Fleming [28], Fleming and Rishel [29], Fleming and
Soner [30] and Lions [32].
When H = H(p) the Lagrangian of our system can be obtained as
L(v) = sup{⟨v, p⟩ − H(p)}.
p
In the case of a smooth convex Hamiltonian the corresponding Lagrangian is strictly convex
but non smooth in general.
26
1.9 Properties of viscosity solutions when H = H(p)
Theorem 1.30. The unique viscosity solution of the Cauchy problem (1.9) is the Lipschitz
continuous function u(t, x) defined for (t, x) ∈ Ω as
{
(
)}
x−y
u(t, x) = min u(0, y) + tL
.
y∈Ω0
t
(1.10)
Theorem 1.31. Let u(t, x) be a viscosity solution of the Cauchy problem (1.9).
i) The minimum point y for (t, x) ∈ Ω in (1.10) is unique if and only if u(t, x) is differentiable
in x. Moreover in this case y = x − tHp (Dx u(t, x)).
ii) (Dynamic Programming Principle) Fix (t, x) ∈ Ω, then for all t′ ∈ [0, t]
{
)}
(
x−z
′
′
u(t, x) = min u(t , z) + (t − t )L
.
z∈Ωt′
t − t′
)
(
iii) Let 0 < s < t, let (t, x) ∈ Ω and y a minimum point in (1.10). Let z = st x + 1 − st y.
Then y is the unique minimum point for
{
(
)}
z−w
u(s, z) = min u(0, y) + sL
.
w∈Ω0
s
Remark 1.32. Note that in this case characteristics are straight lines, for this reason will be
called rays.
Definition 1.33. Let y ∈ Ω0 be a minimizer for u(t, x). We call optimal ray the segment [x, y]
defined in [0, t].
Proposition 1.34. Let [x, y] and [x′ , y ′ ] be two optimal rays in [0, t], for x, x′ ∈ Ωt y, y ′ ∈ Ω0
then they cannot intersect except than at time 0 or t.
Proof. It follows from Theorem 1.31-(iii).
Proposition 1.35. Let u0 be a semiconcave function. Then the unique viscosity solution u(t, x)
of (1.9) is semiconcave in x, for all t ∈ [0, T ].
Proof. See Lemma 3 in Section 3.3 [26].
Theorem 1.36. Suppose H is locally uniformly convex. Then for any t in (0, T ], u(t, ·) is
locally semiconcave with semiconcavity constant C(t) = Ct . Thus for any fixed τ > 0 there exists
a constant C = C(τ ) such that u(t, ·) is semiconcave with constant less than C for any t ≥ τ .
Proof. See Lemma 4 in Section 3.3 [26].
Moreover u is also locally semiconcave in both the variables (t, x) in (0, T ] × Rn .
27
Preliminaries
1.9.1
Duality solutions
We consider a fixed interval of time [0, 1], and we define duality solutions in this time interval.
Definition 1.37. Setting u+ (1, z) := u(1, z), we define duality solutions for s ∈ [0, 1] and
z ∈ Ωs , the backward solution
{
(
)}
x−z
−
+
u (s, z) := max u (1, x) − (1 − s)L
(1.11)
,
x∈Ω1
1−s
and the forward solution
{
(
)}
z−y
−
u (s, z) := min u (0, y) + sL
.
y∈Ω0
s
+
(1.12)
Remark 1.38. Note that the function v(τ, y) := u− (1 − τ, y) is a viscosity solution of
{
∂τ v − H(Dy v) = 0
in Ω ⊂ [0, 1] × Rn
v(0, y) = u(1, y) for all y ∈ Ω1 .
Moreover the forward solution is the viscosity solution of
{
∂t u + H(Dx u) = 0 in Ω ⊂ [0, 1] × Rn ,
u(0, x) = u− (0, x) for all x ∈ Ω0 .
Thanks to the previous remark Theorems 1.31, 1.36 and Propositions 1.34, 1.35 hold for v
and the forward solution u+ .
Proposition 1.39. From the definitions above, u+ and u− satisfy the following properties for
x ∈ Ω1 , y ∈ Ω0 and z ∈ Ωs for s ∈ (0, 1)
u− (1, x) = u+ (1, x) = u(1, x),
u+ (0, y) = u− (0, y) ≤ u(0, y),
u− (s, z) ≤ u+ (s, z) ≤ u(s, z).
Proof. The first two equalities are a consequence of the fact that u0 and u1 , defined as follows,
are L(x − y) conjugate functions. First, for x ∈ Ω1 , set
u1 (x) := min {u(0, y) + L(x − y)},
y∈Ω0
i.e. u1 (x) = u(1, x).
Then, for y ∈ Ω0 , set
u0 (y) := max{u1 (x) − L(x − y)},
x∈Ω1
i.e.
u0 (y)
u− (0, y).
=
From these definitions it follows u0 (y) ≤ u(0, y) and
u1 (x) = min {u0 (y) + L(x − y)}.
y∈Ω0
Indeed, let x̃ ∈ Ω1 a maximizer for u0 (y) then
u0 (y) = u1 (x̃) − L(x̃ − y) ≤ u(0, y) + L(x̃ − y) − L(x̃ − y) = u(0, y).
28
1.9 Properties of viscosity solutions when H = H(p)
Nevertheless, from u0 (y) ≤ u(0, y), it follows
min {u0 (y) + L(x − y)} ≤ min {u(0, y) + L(x − y)} = u1 (x).
y∈Ω0
y∈Ω0
On the other hand, let ỹ be a minimizer for miny∈Ω0 {u0 (y) + L(x − y)}, then we have
min {u0 (y) + L(x − y)} = u0 (ỹ) + L(x − ỹ)
y∈Ω0
≥ u1 (x) − L(x − ỹ) + L(x − ỹ)
= u1 (x).
Note that the definition of u− (s, z) and u+ (s, z) implies that u− (1, x) = u1 (x) and u+ (0, y) =
u0 (y).
The last inequality follows, for s in (0, 1), by
{
(
)}
x−z
u− (s, z) = max u1 (x) − (1 − s)L
x∈Ω1
1−s
{
{
(
)}}
x−z
0
= max min u (y) + L(x − y) − (1 − s)L
x∈Ω1
y∈Ω0
1−s
{
(
)}
z−y
≤ min u0 (y) + sL
= u+ (s, z),
y∈Ω0
s
where the inequality is given by the convexity of L
(
)
(
)
z−y
x−z
L(x − y) ≤ sL
+ (1 − s)L
.
s
1−s
z−y
Note that, from the strict convexity of L, the equality holds if and only if x−z
1−s =
s , i.e.
z = sx + (1 − s)y that is z belongs to the segment joining the maximizer x to the minimizer y.
Furthermore, due to the fact that u− (0, y) ≤ u(0, y), we have u+ (s, z) ≤ u(s, z).
Proposition 1.40. Suppose H is a smooth uniformly convex Hamiltonian. Then a C 1,1 -estimate
holds in the regions where u− (s, z) = u+ (s, z), for s ∈ (0, 1).
Proof. Fix s in (0, 1) and z such that u− (s, z) = u+ (s, z), then as observed in the previous
proof there is a unique segment, connecting the unique minimizer y(z) in (1.12) to the unique
maximizer x(z) in (1.11) and passing through z. Hence z = (1 − s)y(z) + sx(z). Moreover both
u+ (s, ·) and u− (s, ·) are differentiable in z since the minimizer and the maximizer are unique.
Note that neither u− (s, z) = u+ (s, z) implies necessarily that u− (s, z) = u+ (s, z) = u(s, z), nor
u+ (s, z) = u(s, z) implies that u− (s, z) = u+ (s, z) = u(s, z). However, if for a z̃ u− (s, z̃) = u(s, z̃)
then u− (s, z̃) = u+ (s, z̃) = u(s, z̃).
From the definition of u+ and u− for z ′ ∈ Ωt
(
)
( ′
)
x(z) − z ′
z − y(z)
1
−
′
+
′
0
u (x(z)) − (1 − s)L
≤ u (s, z ) ≤ u (s, z ) ≤ u (y(z)) + sL
.
1−s
s
29
Preliminaries
Since z = (1 − s)y(z) + sx(z) and
(
)
(
)
x(z) − z
z − y(z)
0
u (s, z) = u (x(z)) − (1 − s)L
= u (y(z)) + sL
= u+ (s, z),
1−s
s
−
1
we obtain
)
( (
)
(
)
z′ − z
−(1 − s) L x(z) − y(z) −
− L x(z) − y(z) ≤ u− (s, z ′ ) − u− (s, z)
1−s
( (
)
)
(
)
z′ − z
+
′
+
≤ u (s, z ) − u (s, z) ≤ s L x(z) − y(z) +
− L x(z) − y(z) .
s
In particular, recalling the fact that both u+ (s, ·) and u− (s, ·) are differentiable in z, and that
L is C 1
(
)
Dx u− (s, z) = Dx u+ (s, z) = Lv x(z) − y(z) .
Moreover, thanks to the fact that we are considering the region where u+ (s, z) = u− (s, z), they
are both semiconvex and semiconcave in this region, thus we can recover a Lipschitz estimate
for Dx u+ and Dx u− .
−
C
C
|z|2 ≤ u− (s, x+z)+u− (s, x−z)−u− (s, x) = u+ (s, x+z)+u+ (s, x−z)−u+ (s, x) ≤ |z|2 .
1−s
s
Hence we have proved that in the region where u− = u+ the dual solutions are C 1,1 .
Remark 1.41. In the proof of the above proposition we used the semiconcavity of u+ (s, ·) and
the semiconvexity of u− (s, ·) thus the hypothesis of uniform convexity of the Hamiltonian is
necessary.
The definition of backward and forward solutions can be easily generalized for every time
interval [τ, t] ⊂ [0, T ]. Propositions 1.39 and 1.40 hold even in this case.
Definition 1.42. Setting u+
t,τ (t, z) := u(t, z), we define duality solutions for s ∈ [τ, t] and z ∈ Ωs ,
the backward solution
)}
{
(
x−z
−
+
,
ut,τ (s, z) := max ut,τ (t, x) − (t − s)L
x∈Ωt
t−s
and the forward solution
u+
t,τ (s, z)
{
(
)}
z−y
−
:= min ut,τ (τ, y) + (s − τ )L
.
y∈Ωτ
s−τ
30
Chapter 2
SBV Regularity for Hamilton-Jacobi
equations
In this chapter we study the regularity of the viscosity solution of the following Hamilton-Jacobi
equation
∂t u + H(t, x, Dx u) = 0
in Ω ⊂ [0, T ] × Rn .
(2.1)
Under the following assumptions on H
(H1) H ∈ C 3 ([0, T ]×Rn ×Rn ) with bounded second derivatives and there exist positive constants
a, b, c such that
i) H(t, x, p) ≥ −c,
ii) H(t, x, 0) ≤ c,
iii) |Hpx (t, x, p)| ≤ a + b|p|,
(H2) there exists cH > 0 such that
c−1
H Idn (p) ≤ Hpp (t, x, p) ≤ cH Idn (p)
for any t, x,
we will prove the following result.
Theorem 2.1. Let u be a viscosity solution of (2.1), assume (H1), (H2). Then the set of times
S := {t | Dx u(t, ·) ̸∈ [SBVloc (Ωt )]n }
is at most countable. In particular Dx u, ∂t u ∈ [SBVloc (Ω)]n .
Moreover, under the hypotheses
(H1-bis) H ∈ C 3 (Rn ×Rn ) with bounded second derivatives and there exist positive constants a, b, c
such that
i) H(x, p) ≥ −c,
31
SBV Regularity for Hamilton-Jacobi equations
ii) H(x, 0) ≤ c,
iii) |Hpx (x, p)| ≤ a + b|p|,
(H2-bis) there exists cH > 0 such that
c−1
H Idn (p) ≤ Hpp (x, p) ≤ cH Idn (p)
for any x,
as a consequence of the theorem above, the following corollary holds.
Corollary 2.2. Under assumptions (H1 − bis), (H2 − bis), the gradient of any viscosity solution
u of
H(x, Du) = 0
in Ω ⊂ Rn ,
belongs to [SBVloc (Ω)]n .
These results confirm and extend the SBV-regularizing effect of Hamilton-Jacobi equations
which was already proved by Cannarsa, Mennucci and Sinestrari in [19] for strictly convex
Hamiltonians and regular initial data and by Bianchini, De Lellis and Robyr in [9] for uniformly
convex Hamiltonians depending only on Dx u.
The Lipschitzianity of the viscosity solution u in x allows us to conjecture that the SBVregularizing effect is preserved even in the case of a Hamiltonian which depends also on the
solution u,
H(x, u, Du) = 0
in Ω ⊂ Rn .
As we will see, the study of generalized characteristics, which are in general C 2 -curve, and
their approximations with straight lines in small intervals of time, allow us to prove the main
result using a strategy which is standard in showing the SBV-regularizing effect.
The chapter is organized as follows. In Section 2.1 we prove the no-crossing property of
generalized backward characteristics. Finally in Section 2.2 we prove all the necessary lemmas
and Theorem 2.1.
2.1
No-crossing property for a small interval of time
We begin with the study of generalized backward characteristics, defined as in Definition 1.24.
Let t ∈ [0, T ] be fixed, if we restrict to a τ > 0 which is not too far from t, we can establish a one
to one correspondence between generalized backward characteristics and maximizers of (1.8).
Thus we are able to recover regularity and the no-crossing property for generalized backward
1,1 (Ω )
characteristics. Moreover the backward solution u−
s
t,0 (s, ·), defined in (1.8), belongs to C
for every s ∈ (τ, t), while in [0, τ ] it can loose its regularity.
To prove the above fact let us first reduce to a simpler case which will be useful also later
on during the proof of Theorem 2.1.
Lemma 2.3. Consider the solutions to the system
{
˙
ξ(s)
= Hp (s, ξ(s), p(s))
ṗ(s) = −Hx (s, ξ(s), p(s))
32
(2.2)
2.1 No-crossing property for a small interval of time
with final conditions
{
ξ(t) = x
p(t) = p
(2.3)
where x is fixed in Rn and p ∈ K a compact set in Rn . For t − τ small enough there exists a
one to one correspondence between p in K and ξ(τ ) when ξ(·) is a solution of (2.2),(2.3).
Proof. Thanks to the Taylor expansion of the flow generated by (2.2), the solution to that
system, with (2.3) as final conditions, is equal to
ξ(τ ) = x − (t − τ )Hp (t, x, p) + O((t − τ )2 ),
and differentiating in p
ξp (τ ) = −(t − τ )Hpp (t, x, p) + O((t − τ )2 ).
(2.4)
Note that ξp and pp satisfy
{
ξ˙p (s) = Hpx (s, ξ(s), p(s))ξp (s) + Hpp (s, ξ(s), p(s))pp (s)
ṗp (s) = −Hxx (s, ξ(s), p(s))ξp (s) − Hxp (s, ξ(s), p(s))pp (s)
with final conditions
{
ξp (t) = 0
pp (t) = Idn (p).
Since they are smooth, equation (2.4) is precisely the Taylor expansion of ξp (τ ).
)
Call ω := x−ξ(τ
t−τ . Last equation implies that ωp is uniformly different from zero since
ωp = Hpp (t, x, p) + O(t − τ ).
Thus, restricting to t − τ small enough, we can locally invert this equation and obtain
pω = Lvv (t, x, ω) + O(t − τ ).
(2.5)
Moreover, from
ω = Hp (t, x, p) + O(t − τ ),
integrating (2.5), we obtain
p = Lv (t, x, ω) + O(t − τ ).
Thus we have reached a one to one correspondence between ξ(τ ) and the value p of its dual
curve at time t.
Integrating (2.4) in p between p1 and p2 we obtain
ξ1 (τ ) − ξ2 (τ )
= Hp (t, x, p1 ) − Hp (t, x, p2 ) + O(t − τ )(p1 − p2 )
τ −t
where ξ1 and ξ2 are the generalized backward characteristics with initial data p1 and p2 respectively.
33
SBV Regularity for Hamilton-Jacobi equations
Proposition 2.4. Consider a solution ξ to the system (2.2) with final conditions (2.3), let
y := ξ(τ ) and consider the straight line joining x to y
η(s) =
s−τ
t−s
x+
y.
t−τ
t−τ
Then we have the following estimates
∥η − ξ∥[C 0 ([τ,t])]n , ∥ηp − ξp ∥[C 0 ([τ,t])]n2 , ∥ηpp − ξpp ∥[C 0 ([τ,t])]n3 ≤ O((t − τ )2 ),
˙ [C 0 ([τ,t])]n , ∥η̇p − ξ˙p ∥ 0
∥η̇ − ξ∥
, ∥η̇pp − ξ˙pp ∥[C 0 ([τ,t])]n3 ≤ O(t − τ ).
[C ([τ,t])]n2
Proof. As we saw in the previous proposition
y = ξ(τ ) = x − (t − τ )Hp (t, x, p) + O((t − τ )2 ),
and for s ∈ [τ, t]
ξ(s) = x − (t − s)Hp (t, x, p) + O((t − s)2 ).
Compute now the difference
s − τ
t−s
2 sup |η(s) − ξ(s)| = sup x+
y − x + (t − s)Hp (t, x, p) + O((t − s) )
t−τ
s∈[τ,t]
s∈[τ,t] t − τ
t − s (
) t−s
x − (t − τ )Hp (t, x, p) + O((t − τ )2 ) −
x
= sup t−τ
s∈[τ,t] t − τ
2 + (t − s)Hp (t, x, p) + O((t − s) )
≤ O((t − τ )2 ).
Moreover from
yp = ξp (τ ) = −(t − τ )Hpp (t, x, p) + O((t − τ )2 ),
and from
ξp (s) = −(t − s)Hpp (t, x, p) + O((t − s)2 )
for s ∈ [τ, t], we obtain
t − s
2 sup |ηp (s) − ξp (s)| = sup yp + (t − s)Hpp (t, x, p) + O((t − s) )
s∈[τ,t]
s∈[τ,t] t − τ
t − s (
)
= sup −(t − τ )Hpp (t, x, p) + O((t − τ )2 )
s∈[τ,t] t − τ
2 + (t − s)Hpp (t, x, p) + O((t − s) )
≤ O((t − τ )2 ).
In an analogous way, from
ypp = ξpp (τ ) = −(t − τ )Hppp (t, x, p) + O((t − τ )2 ),
34
(2.6)
2.1 No-crossing property for a small interval of time
and from
ξpp (s) = −(t − s)Hppp (t, x, p) + O((t − s)2 )
for s ∈ [τ, t], we obtain
sup |ηpp (s) − ξpp (s)| ≤ O((t − τ )2 ).
s∈[τ,t]
Observe now that
η̇(s) =
x−y
,
t−τ
and
˙
ξ(s)
= −Hp (t, x, p) + O(t − s),
hence
x − y
˙
−
H
(t,
x,
p)
+
O((t
−
s))
sup |η̇(s) − ξ(s)| = sup p
s∈[τ,t]
s∈[τ,t] t − τ
= sup Hp (t, x, p) + O(t − τ ) − Hp (t, x, p) + O(t − s)
s∈[τ,t]
≤ O(t − τ ).
In the same way we obtain
sup |η̇p (s) − ξ˙p (s)| ≤ O(t − τ ),
s∈[τ,t]
and
sup |η̇pp (s) − ξ˙pp (s)| ≤ O(t − τ ).
s∈[τ,t]
Now, fix x ∈ Rn and a compact set K ⊂ Rn . Call ξ(τ, K) the subset of Rn defined as
ξ(τ, K) := {ξ(τ )| ξ is a solution of (2.2) with final conditions (2.3)}.
For any y in ξ(τ, K) consider the function
{∫ t
˙
ϕ(τ, y, t, x) := min
L(s, ξ(s), ξ(s))ds
τ
}
2
n
ξ
∈
[C
([τ,
t])]
,
ξ(τ
)
=
y,
ξ(t)
=
x,
,
and observe that for any y ∈ ξ(τ, K) there exists a unique ξ solution of (2.2) with final conditions
(2.3) such that y = ξ(τ, p). Thus we can see y as y = y(p) with a C 2 dependence of y from p.
Proposition 2.5. It holds
(
)
ϕ(τ, y(p), t, x) − (t − τ )L t, x, x − y(p) ≤ O((t − τ )2 ).
2
t−τ
C (K)
In particular for t−τ small enough y 7→ ϕ(τ, y, t, x) and x 7→ ϕ(τ, y, t, x) are convex with constant
C̃
t−τ .
35
SBV Regularity for Hamilton-Jacobi equations
Proof. Note that, from the definition, y 7→ ϕ(τ, y, t, x) and and x 7→ ϕ(τ, y, t, x) are automatically
semiconvex.
Moreover, it is enough to consider the function y 7→ ϕ(τ, y, t, x) since there is a symmetry
between y 7→ ϕ(τ, y, t, x) and x 7→ ϕ(τ, y, t, x). Thus the analysis of the two functions is similar.
From the definition, the function y 7→ ϕ(τ, y, t, x) has a unique minimum ξ which is the
solution to system (2.2) with final conditions (2.3). Thus the C 2 dependence of y from p
implies, for a small t − τ , that p 7→ ϕ(τ, y(p), t, x) belongs to C 2 (K).
Let ξ be the unique minimizer for ϕ(τ, y, t, x) and observe that x = η(t) and
where η is the straight line joining x to y as in (2.6).
x−y
t−τ
(
)
x − y(p) sup ϕ(τ, y(p), t, x) − (t − τ )L t, x,
=
t−τ
p∈K
∫ t
∫ t
˙
L(t, η(t), η̇(t))ds
= sup L(s, ξ(s), ξ(s))ds
−
p∈K
τ
τ
{ ∫
∫ t
t
˙
˙
≤ sup L(s, ξ(s), ξ(s))ds −
L(t, ξ(s), ξ(s))ds
p∈K
τ
τ
∫ t
∫ t
˙
˙
L(t, η(t), ξ(s))ds
+
L(t, ξ(s), ξ(s))ds −
τ
τ
}
∫ t
∫ t
˙
L(t, η(t), η̇(t))ds
+ L(t, η(t), ξ(s))ds
−
τ
τ
}
{ ∫ t
∫ t
∫ t
˙
|ξ(s) − η̇(t)|ds
|ξ(s) − η(t)|ds + C3
|s − t|ds + C2
≤ sup C1
p∈K
≤ sup
p∈K
τ
τ
{
∫
τ
t
C1
(t − τ )2 + C2 Hp (t, x, p)
|t − s|ds
2
τ
}
∫ t
∫ t
2
+ C2
O((t − s) )ds + C3
O((t − s))ds
−
τ
τ
≤ O((t − s) ).
2
Moreover for the first derivative
supp∈K
=
[
(
)]
∂p ϕ(τ, y(p), t, x) − (t − τ )L t, x, x − y(p) =
t−τ
∫ t
∫ t
˙
˙
Lx (s, ξ(s), ξ(s))ξ
Lv (s, ξ(s), ξ(s))
ξ˙p (s)ds
sup p (s)ds +
p∈K
τ
τ
∫
t
−
∫
Lx (t, η(t), η̇(t))ηp (t)ds −
τ
τ
36
t
Lv (t, η(t), η̇(t))η̇p (t)ds
= η̇(t),
2.1 No-crossing property for a small interval of time
{ ∫
t
2
˙
≤ sup Lx (s, ξ(s), ξ(s))(−(t
− s)Hp (t, x, p) + O(t − s) )ds
p∈K
≤ sup
p∈K
τ
∫ t
}
˙
+ (Lv (s, ξ(s), ξ(s))
− Lv (t, η(t), η̇(t)))(−Hpp (t, x, p) + O(t − τ ))ds
τ
{ ∫
t
C1
|(s − t) + O(t − s)|ds
τ
∫ t
}
˙
+ [Lv (s, ξ(s), ξ(s))
− Lv (t, η(t), η̇(t))](C2 + O(t − τ ))ds
τ
≤ O((t − s) ).
2
Analogously for the second derivative
)]
[
(
x
−
y(p)
≤ O((t − s)2 ).
sup ∂pp ϕ(τ, y(p), t, x) − (t − τ )L t, x,
t
−
τ
p∈K
The map p 7→ y(p) is C 2 (K), it has bounded derivative and the same holds
( true also
) for its
x−y
inverse, due to Proposition 2.4. Thus it follows that ϕ(τ, y, t, x) and (t − τ )L t, x, t−τ are close
in C 2 (K̃), where K̃ is the image of K through the map p 7→ y(p).
Therefore
y 7→ ϕ(τ, y, t, x) is
(
)
x−y
C̃
convex with constant t−τ , the same constant of y 7→ (t − τ )L t, x, t−τ .
Remark 2.6. All the estimates found strictly depend on the compact set K, however thanks to
the finite speed of propagation of the minimizers ξ, see point (iii) of Theorem 1.19, the estimates
can be made uniform for our u−
t,0 .
Let us now come back to our case.
Proposition 2.7. For 0 ≤ τ < t consider the backward solution defined in (1.8) for y in Ωτ .
Then for t − τ small enough the maximum is unique for all y ∈ Ωτ .
Proof. The backward solution can be written in this equivalent way
u−
t,0 (τ, y) = max {u(t, x) − ϕ(τ, y, t, x)} .
x∈Ωt
Recalling that u(t, ·) is semiconcave with constant
with constant
C̃
t−τ ,
C
t
(2.7)
and that −ϕ(τ, y, t, ·) is strictly concave
we can rewrite (2.7) as
}
{
C 2
C 2
−
ut,0 (τ, y) = max u(t, x) − |x| − ϕ(τ, y, t, x) + |x| .
x∈Ωt
t
t
Hence, since u(t, x) − Ct |x|2 is concave and −ϕ(τ, y, t, x) + Ct |x|2 remains strictly concave, the
function u−
t,0 (τ, y) is the maximum of a strictly concave function, hence this maximum is unique.
Thus there exists a unique x ∈ Ωt such that
u−
t,0 (τ, y) = u(t, x) − ϕ(τ, y, t, x),
37
SBV Regularity for Hamilton-Jacobi equations
i.e. there exists a unique curve ξ ∈ [C 2 ([τ, t])]n such that ξ(τ ) = y, ξ(t) = x and
∫ t
−
˙
L(s, ξ(s), ξ(s))ds.
ut,0 (τ, y) = u(t, ξ(t)) −
τ
1,1 (Ω ).
Corollary 2.8. For t − τ small enough and s ∈ (τ, t) the function u−
s
t,0 (τ, ·) is C
1
Proof. From the above proposition we know that u−
t,0 (s, ·) is C (Ωs ) for every s ∈ [τ, t). Consider
now the forward solution defined from u−
t,0 (τ, ·)
{
}
∫ s
+
−
2
n
˙
ut,τ (s, x) := min ut,0 (τ, ξ(τ )) +
L(l, ξ(l), ξ(l))dl
ξ(s) = x, ξ ∈ [C ([τ, s])] .
τ
u−
t,0 (τ, y)
Due to the fact that
has a unique maximizer for every y ∈ Ωτ we have that u+
t,τ (s, x) =
−
−
ut,0 (s, x) for every s ∈ [τ, t] and x ∈ Ωs . Thus for s ∈ (τ, t), ut,0 (s, ·) is both semiconvex and
semiconcave, hence C 1,1 (Ωs ).
Remark 2.9. As a consequence of Proposition 2.7, for every y ∈ Ωτ there exists only one curve
which is a maximizer for the function ũ(τ, y) and a generalized backward characteristic. Hence
generalized backward characteristics which are also maximizers do not intersect even at time τ .
It remains to prove the following.
Proposition 2.10. Every generalized backward characteristic ξ(s), i.e. a solution of (2.2) with
final conditions (2.3) where p ∈ Dx+ u(t, x), is a maximizer for u−
t,0 (τ, ξ(τ )) if t − τ is small
enough.
Proof. Let ξ be a generalized backward characteristic with ξ(t) = x, p(t) = p ∈ Dx+ u(t, x) and
ξ(τ ) = y. Then ξ is a minimizer for ϕ(τ, t, y, x) and p = p(t) = −Dy ϕ(τ, y, t, x).
˜
Let ξ˜ be the unique maximizer for u−
t,0 (τ, y) and suppose by contradiction that ξ differs from
˜ = x̃ ̸= x = ξ(t). Then by definition
ξ, in particular ξ(t)
u−
t,0 (τ, y) = u(t, x̃) − ϕ(τ, y, t, x̃) > u(t, x) − ϕ(τ, y, t, x).
Thus, for the differentiability and the convexity of ϕ(τ, y, t, ·)
u(t, x̃) − u(t, x) > ϕ(τ, y, t, x̃) − ϕ(τ, y, t, x)
≥ ⟨Dy ϕ(τ, y, t, x), x̃ − x⟩ +
C̃
|x̃ − x|2 .
t−τ
On the other hand for the semiconcavity of u(t, ·)
u(t, x̃) − u(t, x) < ⟨p, x̃ − x⟩ +
C
|x̃ − x|2 .
t
Recalling that p = −Dy ϕ(τ, y, t, x), for t − τ small enough we reach the absurd
C̃
C
>
.
t
t−τ
38
2.2 Proof of Theorem 2.1
From the above proposition it follows
Corollary 2.11. Generalized backward characteristics cannot intersect in [τ, t) if t − τ is small
enough.
2.2
2.2.1
Proof of Theorem 2.1
Preliminary remarks
Let u be a viscosity solution of (2.1). Applying Proposition 1.18 we can assume without loss
of generality that u is a solution of the Cauchy Problem (2.1) with a bounded Lipschitz initial
datum u(0, x) = u0 (x) over a bounded domain [0, δ] × U. Moreover assumptions (H1)-(H2)
guarantee that the Hamiltonian is convex and has super-linear growth in the last variable.
We will prove the SBV regularity over the smaller interval of time [τ, τ + ε] for a fixed τ > 0,
ε > 0 small enough and such that [τ, τ + ε] ⊂ [0, δ]. As we have already seen, this is necessary
to prevent intersections of generalized backward characteristics.
We consider a ball BR (0) ⊂ Rn and a bounded convex set Ω ⊂ [τ, τ + ε] × Rn with the
properties that
• {s} × BR (0) ⊂ Ω for every s ∈ [τ, τ + ε];
• for any (t, x) ∈ Ω and for any C 2 curve ξ which minimizes u(t, x) in (1.3), the entire curve
ξ(s) for s ∈ [τ, t] is contained in Ω.
Indeed, from the fact that ∥Du∥∞ < ∞, it is enough to choose
{
}
Ω := (t, x) ∈ [τ, τ + ε] × Rn | |x| ≤ R + C ′ (τ + ε − t)
with C ′ sufficiently large and depending only on ∥Du∥∞ and H.
The general idea of the proof is now standard, see [4], [9]. We construct a monotone bounded
functional F (t) defined on the interval [τ, τ + ε]. Then, we relate the presence of a Cantor part
in the matrix Dx2 u(t, ·) for a certain t in [τ, τ + ε] with a jump of the functional F in t. Since
this functional can have only a countable number of jumps, the Cantor part of Dx2 u(t, ·) can be
different from zero only for a countable number of t’s.
Remark 2.12. Once we have formalized the above strategy and proved the SBV regularity for
almost every t in [τ, τ + ε] the conclusion that Dx u belongs to [SBVloc (Ω)]n follows from the
slicing theory of BV functions (see Theorem 3.108 of [5]). The local SBV regularity of ∂t u
follows instead from the Volpert chain rule.
2.2.2
Construction of the functional F
Consider t belonging to (τ, τ + ε] for a fixed τ > 0 and ε > 0 small enough. For any τ ≤ s < t
we define the set-valued map
Xt,s (x) := {ξ(s)| ξ(·) is a solution of (2.2), with ξ(t) = x, p(t) = p ∈ Dx+ u(t, x)}.
39
SBV Regularity for Hamilton-Jacobi equations
Moreover we will denote by χt,s the restriction of Xt,s to the points where it is single-valued.
According to Theorem 1.25, the domain of χt,s , dom(χt,s ) =: Ut , consists of those points where
Dx+ u(t, x) is single-valued, i.e. there exists a unique minimizer for u(t, x). For that reason χt,s
is clearly defined Hn -a.e. in Ωt . We will sometimes write χt,s (Ωt ) meaning χt,s (Ut ).
Remark 2.13. In the definition of Xt,s we follow generalized backward characteristics starting
at time t > 0 till time s. As we have already seen, if t − s is small enough, generalized backward
characteristics cannot intersect except than at time t. Thus if we choose ε > 0 small enough we
have the injectivity of the set valued map Xt,τ over the interval of time [τ, τ + ε].
Note that in the case H = H(Dx u) the authors of [9] were able, in Proposition 5.2, to prove
the injectivity of Xt,0 , as a set-valued map, for every t ∈ [0, ε] with ε small enough.
Therefore, equivalently to Proposition 5.2 in [9], we can state
Proposition 2.14. Let t be fixed such that τ < t ≤ τ + ε, for an ε > 0 small enough, which
does not depend on t. Then taken any two solutions (ξ1 , p1 ) and (ξ2 , p2 ) of the system (2.2) with
final condition
ξi (t) = xi ∈ Ωt
pi (t) ∈ Dx+ u(t, xi )
i = 1, 2,
and (ξ1 (t), p1 (t)) ̸= (ξ2 (t), p2 (t)) it follows that ξ1 (τ ) ̸= ξ2 (τ ). Hence, in particular, the map
x 7→ Xt,τ (x) is injective as a set-valued map.
Proof. It follows from Corollary 2.11.
For every τ < t ≤ τ + ε, we can now define the functional
F (t) := Hn (χt,τ (Ut )).
(2.8)
Lemma 2.15. The functional F is non increasing,
F (s) ≥ F (t)
for any s, t ∈ (τ, τ + ε] with s < t.
Proof. As in the proof of Lemma 4.1 in [9], the claim follows from the following consideration:
χt,τ (Ωt ) ⊂ χs,τ (Ωs )
for every τ ≤ s ≤ t ≤ τ + ε.
Indeed, consider any y ∈ χt,τ (Ωt ). Then there exists a curve ξ in [C 2 ([τ, t])]n and a point
x ∈ Ωt such that ξ is the unique minimizer in (1.3) with the following endpoints conditions
ξ(t) = x, ξ(τ ) = y. Such a curve remains the unique minimizer also for u(s, ξ(s)) for any
τ ≤ s ≤ t ≤ τ + ε. Hence, setting z = ξ(s), we have that the point y can be seen as y = χs,τ (z)
and y ∈ χs,τ (Ωs ).
2.2.3
Hille-Yosida transformation
Take a Borel set A ⊂ Ωt for a fixed time t ∈ (τ, τ + ε]. In order to compute the measure
Hn (Xt,τ (A)) we follow the evolution of the set along generalized backward characteristics till
the time τ .
40
2.2 Proof of Theorem 2.1
Let us recall how the characteristics and their dual arc evolve in time. They are solutions of
the system (2.2), together with the final condition (2.3) where p belongs to Dx+ u(t, x).
We have to face the following problem: the function Dx+ u(t, ·) is a multi-valued function
of bounded variation which is not Lipschitz in general. However it can be easily related to a
maximal monotone function whose graph can be parametrized in a Lipschitz way as shown in
Alberti and Ambrosio [1].
Let us consider the graph (A, Dx+ u(t, A)) for a Borel set A ⊂ Ωt . Since u(t, x) is semiconcave
in x, v(x) := −(u(t, x) − 12 C|x|2 ) is a convex function. Note that the semiconcavity constant
should depend on t, i.e. C(t) = Ct , however a uniform one can be taken due to the fact that
t belongs to (τ, τ + ε] where τ > 0. Moreover, as seen in Theorem 1.14-(iv), the differential
of v is a maximal monotone function. It can be proven, see for example [1], that the graph
of a maximal monotone function is a Lipschitz submanifold without boundary. Adapting the
same procedure to our case, we can parametrize the graph of the derivative of our semiconcave
function with a 1-Lipschitz function.
Indeed, we pass from our graph {(x, Dx+ u(t, x))| x ∈ A} to the graph of a maximal monotone
function with the following transformation
{
x=x
y = Cx − p,
where C is the semiconcavity constant of u(t, ·). Then we apply an Hille-Yosida transformation
to have a 1-Lipschitz parametrization of it.
{
z =x+y
w = y.
Call T (x) := Dx v(x) the maximal monotone function. Retracing the passages above, we can
express w as a 1-Lipschitz single-valued function of z. Taking z ∈ B := A + T (A)
{
z=z
w = (Idn + (T )−1 )−1 (z).
Thus, coming back to our original coordinates, we can describe our graph with the following
Lipschitz parametrization
{
x(z) = z − w(z)
(2.9)
p(z) = Cz − (C + 1)w(z),
where z ∈ B, i.e. we have
ΓA := {(x, Dx+ u(t, x))| x ∈ A} = {(z − w(z), Cz − (C + 1)w(z))| z ∈ B}.
Remark 2.16. As explained in [1] the 1-Lipschitz function w(z) is exactly the derivative of the
inf-convolution function of v(x) = −(u(t, x) − 12 C|x|2 )
}
{
|x − z|2
.
f (z) = minn v(x) +
x∈R
2
Thus we have w(z) = fz (z) where f is a convex function.
41
SBV Regularity for Hamilton-Jacobi equations
When applying the flux backward in time, starting from our set ΓA , characteristics ξ(s, z)
and p(s, z) evolve according to
{
with final conditions
{
˙ z) = Hp (s, ξ(s, z), p(s, z))
ξ(s,
ṗ(s, z) = −Hx (s, ξ(s, z), p(s, z))
(2.10)
ξ(t, z) = x(z) = z − w(z)
p(t, z) = p(z) = Cz − (C + 1)w(z),
(2.11)
for z in B. Since the flux is described by smooth equations and thanks to the fact that the
parametrization of our initial set is 1-Lipschitz, the solutions ξ(s, z), p(s, z) are Lipschitz curves.
We can now rewrite Xt,τ in an equivalent way, for x in A
Xt,τ (x) = {ξ(τ ) | ξ(·) is a solution of (2.2), with ξ(t) = x, p(t) = p ∈ Dx+ u(t, x)}
= {ξ(τ, z) | ξ(·, z) is a solution of (2.10), with ξ(t, z) = z − w(z),
p(t, z) = Cz − (C + 1)w(z), z ∈ x + T (x)}.
With an abuse of notation we will denote with ξ(τ, ·) : B → Ωτ the function Xt,τ (·) when we are
considering the Lipschitz parametrization; with this notation Xt,τ (A) = ξ(τ, B). We can now
apply the Area Formula to ξ(τ, ·)
∫
H ((ξ(τ, ·)
0
−1
∫
| det(ξz (τ, z))|dz.
(w))dw =
ξ(τ,B)
(2.12)
B
Thanks to the injectivity of the map Xt,τ which is preserved when passing to the Lipschitz
parametrization, the left term of (2.12) is precisely the measure of the set ξ(τ, B).
Hence, we have
∫
H0 ((ξ(τ, ·)−1 (w))dw = Hn (ξ(τ, B)) = Hn (Xt,τ (A)).
ξ(τ,B)
To compute det(ξz (τ, z)) we differentiate in z the equations (2.10), (2.11) obtaining that ξz
and pz satisfy the system
{
ξ˙z (s, z) = Hpx (s, ξ(s, z), p(s, z))ξz (s, z) + Hpp (s, ξ(s, z), p(s, z))pz (s, z)
ṗz (s, z) = −Hxx (s, ξ(s, z), p(s, z))ξz (s, z) − Hxp (s, ξ(s, z), p(s, z))pz (s, z)
(2.13)
with the final conditions
{
ξz (t, z) = Idn (z) − wz (z)
pz (t, z) = CIdn (z) − (C + 1)wz (z),
for any z ∈ B.
42
(2.14)
2.2 Proof of Theorem 2.1
2.2.4
Approximation and area estimates
If we choose ε > 0 small enough we can approximate our curves with straight lines for any t in
(τ, τ + ε], i.e. we can write
˙ z) + O((t − τ )2 ).
ξ(τ, z) = ξ(t, z) − (t − τ )ξ(t,
Using this approximation and (2.13) we obtain
(
)
det(ξz (τ, z)) = det ξz (t, z) − (t − τ )Hpx (t, x(z), p(z))ξz (t, z) − (t − τ )Hpp (t, x(z), p(z))pz (t, z)
+O((t − τ )2 ).
(2.15)
Since we are now considering nearly straight lines, instead of more general curves, we can expect
that this approximation should allow us to adapt the techniques of [9] and recover the lemmas
needed.
Before going on, let us give an explicit formula for the spatial Laplacian of our solution.
Thanks to the semiconcavity of u(t, ·) its spatial Laplacian is a measure. Moreover, using the
1-Lipschitz parametrization given by Hille-Yosida, the spatial Laplacian can be seen as the
push-forward of a particular measure.
Lemma 2.17. For any Borel set A, let {(x(z), p(z))| z ∈ A+T (A)} be the 1-Lipschitz parametrization of the set {(x, Dx+ u(t, x)| x ∈ A} as seen above in (2.9). Then we have



∑ ∂pi (z)
∆u(t, A) = x(z)♯ 
[cof xz (z)]ik  Hn  (A).
∂zk
i,k
Here cof A is the cofactor matrix of the matrix A.
This formula has been shown to the authors by C. De Lellis.
Proof. We can assume A open. Take any ϕ in Cc∞ (Rn ) and compute
∫
∫
∂ϕ(x)
2
ϕ(x)d[Dx u(t, x)]ij = − [Dx u(t, x)]i
dx
∂xj
A
A
∫
∂ϕ(x(z))
=−
pi (z)
det(xz (z))dz
∂xj
A+T (A)
∫
∑ ( ∂ϕ(x(z)) ∂zk (x(z)) )
=−
pi (z)
det(xz (z))dz
∂zk
∂xj
A+T (A)
k
)
∫
∑ ( ∂ϕ(x(z))
=−
pi (z)
[cof xz (z)]jk dz
∂zk
A+T (A)
k
)
∫
∑ ( ∂pi (z)
=
ϕ(x(z))
[cof xz (z)]jk dz
∂zk
A+T (A)
k
)
∫
∑( ∂
[cof xz (z)]jk dz.
+
ϕ(x(z))pi (z)
∂zk
A+T (A)
k
43
SBV Regularity for Hamilton-Jacobi equations
In the lines above we have used the 1-Lipschitz parametrization of the set {(x, Dx+ u(t, x))| x ∈ A}
and the fact that
∂zk (x(z))
1
= [xz (z)]−1
=
[cof xz (z)]jk .
kj
∂xj
det(xz (z))
Now, repeating upside down the passages starting from the last term, one obtains that
)
∫
∑( ∂
ϕ(x(z))pi (z)
[cof xz (z)]jk dz
∂zk
A+T (A)
k
)
∫
)
∑( ∂ (
=−
ϕ(x(z))pi (z) [cof xz (z)]jk dz
∂zk
A+T (A) k
)
∫
) ∂z (x(z))
∑( ∂ (
k
det(xz (z)) dz
=−
ϕ(x(z))pi (z)
∂zk
∂xj
A+T (A) k
∫
)
∂ (
=−
ϕ(x(z))pi (z) det(xz (z))dz
A+T (A) ∂xj
∫
)
∂ (
=−
ϕ(x)[Dx u(t, x)]i dx
A ∂xj
which is equal to zero due to the fact that ϕ has compact support. Hence
)
∑( ∂
[cof xz (z)]jk = 0.
∂zk
k
We are now able to prove an analogous of Lemma 4.3 in [9].
Lemma 2.18. For ε small enough (depending only on the bound M for ∥Hpx ∥), let t ∈ (τ, τ + ε]
and A ⊂ Ωt be a Borel set. Then
∫
n
n
H (Xt,τ (A)) ≥ C1 H (A) − C2 (t − τ )
d∆u(t, ·) + O((t − τ )2 ),
A
where C1 , C2 are positive constants (depending on C, cH ). ∆u(t, ·) is the spatial Laplacian of
u(t, ·).
Proof. Let us start from (2.15).
For t − τ small enough the matrix
Idn (z) − (t − τ )Hpx (t, x(z), p(z))
is invertible. Indeed, since ∃M > 0 such that the norm ∥Hpx (·, ·, ·)∥ < M it is sufficient to take
1
ε < 2nM
. This condition ensures that
det(Idn (z) − (t − τ )Hpx (t, x(z), p(z))) >
44
1
> 0.
2
2.2 Proof of Theorem 2.1
Thus this determinant can be put in evidence in (2.15)
(
)
| det(ξz (τ, z))| = | det (Idn − (t − τ )Hpx ) || det ξz − (t − τ )(Idn − (t − τ )Hpx )−1 Hpp pz | + O((t − τ )2 )
1
> | det (ξz − (t − τ )Hpp pz ) | + O((t − τ )2 ).
2
To lighten the computation above we have omitted the dependence of Hpx , Hpp from t, x(z), p(z)
and of ξz , pz from t, z. Moreover we used the fact that for t − τ small enough it is possible to
expand the inverse
(Idn − (t − τ )Hpx )−1 = Idn + (t − τ )Hpx + O((t − τ )2 ).
We are then left to expand the determinant in series
)
(
det (ξz − (t − τ )Hpp pz ) = det (ξz ) − (t − τ )tr [cof ξz ]T Hpp pz + O((t − τ )2 ),
and use that w = fz as underlined in the Remark 2.16, so that, recalling (2.14),
ξz = Idn − wz = Idn − fzz ,
pz = CIdn − (C + 1)wz = CIdn − (C + 1)fzz .
Call λi , for i = 1, ..., n, the eigenvalues of the positive semidefinite matrix fzz . Hence we can
compute
∏
∏
det (ξz ) =
(1 − λi )
[cof ξz ]ii =
(1 − λj ).
i
j̸=i
The convexity of f and the 1-Lipschitzianity of fz imply that all the eigenvalues are bounded
from above and from below: 0 ≤ λi ≤ 1, for i = 1, . . . , n. Thus, for every i = 1, . . . , n, we have
0 ≤ 1 − λi ≤ 1 and −1 ≤ C − (C + 1)λi ≤ C, in particular this last inequality suggests that we
have to work a bit to bound our determinant, since C − (C + 1)λi has no definite sign.
(
))
1(
det (ξz ) − (t − τ )tr [cof ξz ]T Hpp pz + O((t − τ )2 ) =
2




∏
∏
1
=  (1 − λi ) − (t − τ )tr diag  (1 − λj ) Hpp diag[C − (C + 1)λi ] + O((t − τ )2 )
2
i
j̸=i


∏
∑
∏
1
=  (1 − λi ) − (t − τ )
(1 − λj )[Hpp ]ii (C − (C + 1)λi ) + O((t − τ )2 )
2
i
i
j̸=i
i
j̸=i
1∏
1 ∑∏
=
(1 − λi ) − (t − τ )
(1 − λj )[Hpp ]ii (C(1 − λi ) − λi ) + O((t − τ )2 )
2
2
i
1
= (1 − (t − τ )C tr Hpp )
2
∏
i
(1 − λi ) + (t − τ )
∏
1∑
λi [Hpp ]ii (1 − λj ) + O((t − τ )2 ).
2
i
j̸=i
Now that all the terms have positive sign for an ε small enough, we can use the uniform convexity
of H in p and the bounds on λi to show that there exist constants C1 , C2 , all of them depending
45
SBV Regularity for Hamilton-Jacobi equations
only on C, cH , such that
∏
∑ ∏
| det(ξz (τ, z))| ≥ C1 (1 − λi ) + (t − τ )C2
λi (1 − λj ) + O((t − τ )2 )
i
≥ C1
∏
i
(1 − λi ) + (t − τ )C2
∑
i
i
j̸=i
λi
∏
(1 − λj ) − n(t − τ )C2 C
∏
(1 − λj )
i
j̸=i
+O((t − τ )2 )
∏
∑
∏
= C1 (1 − λi ) − (t − τ )C2
(C(1 − λi ) − λi )) (1 − λj ) + O((t − τ )2 )
i
i
= C1
∏
(1 − λi ) − (t − τ )C2
i
∑
j̸=i
(C − (C + 1)λi )
i
∏
(1 − λj ) + O((t − τ )2 ).
j̸=i
Therefore if we compute the area formula (2.12) we obtain


∫
∫
∏
∑
∏
C1 (1 − λi ) − (t − τ )C2
| det(ξz (τ, z))|dz ≥
(C − (C + 1)λi ) (1 − λj ) dz
B
B
i
i
j̸=i
+O((t − τ ) ).
2
Applying Lemma (2.17) and recalling that 1 − λi are the eigenvalues of ξz (t, z) we obtain the
thesis.
∫
n
n
H (Xt,τ (A)) ≥ C1 H (A) − C2 (t − τ )
d∆u(t, ·) + O((t − τ )2 )
A
where C1 , C2 are constants depending only on C, cH .
In order to complete the proof of the main theorem we need to prove a Lemma which states
the equivalent result of Lemma 5.1 in [9].
Lemma 2.19. If ε > 0 is small enough, for any t ∈ (τ, τ + ε], any δ ∈ [0, t − τ ] and any Borel
set A ⊂ Ωt we have
( )n (
)
1
t − (τ + δ) n n
n
H (Xt,τ +δ (A)) ≥
H (Xt,τ (A)).
2
t−τ
Proof. Fix t in (τ, τ + ε], and let A be a Borel set A ⊂ Ωt . Without loss of generality we can
suppose A to be a compact set.
Consider an approximation of the vector field induced by our generalized backward characteristics by taking a dense sequence of points {xi }∞
i=1 in A. Fix an integer I > 0, call
AI := {xi | i = 1, . . . , I} and define for any s such that τ ≤ s < t and y ∈ Xt,s (A)
(uI )−
t,= (s, y)
}
{
∫ t
2
˙
:= max u(t, ξ(t)) −
L(l, ξ(l), ξ(l))dl ξ is a C ([s, t]) curve, ξ(s) = y, ξ(t) ∈ AI .
s
We assume in addition that the sequence {xi }i∈I is big enough so that we can uniformly
bound the speed of propagation of every maximizer ξ.
46
2.2 Proof of Theorem 2.1
Remark 2.20. All the properties which we stated for maximizers of the backward solution and
for the backward solution itself are preserved in each cone of propagation for the maximizers
of this approximated backward solution (Euler equation, systems for maximizer and dual arc,
no-crossing property, etc) and for (uI )−
t,0 (a.e. differentiability, dynamic programming principle,
semiconvexity).
Through this approximation the set Es := Xt,s (A) is split into at most I open regions Esi ,
i = 1, . . . , I, defined by
Esi := interior of {y ∈ Xt,s (A)| ∃ξ maximizer for (uI )−
t,0 (s, y) such that ξ(t) = xi },
together with the set
JsI :=
)
∪( i
j
Es ∩ Es
i̸=j
of negligible Hn -measure. Indeed, even for (uI )−
t,0 (s, ·) the set of points with more than one
maximum is the set of point of non differentiability and this set has Hn -measure zero.
Call
i
I
Xt,s
(xi ) := {ξ(s)| ξ is a maximizer for (uI )−
t,0 (s, y) with y ∈ E s },
this is a multi-valued function defined on the set AI .
I (A ) converges in the Hausdorff sense to the set X (A) as I tends to infinity.
The set Xt,s
t,s
I
Indeed, it follows from the strong convergence of the maximizers of (uI )−
t,0 to the maximizers of
−
ut,δ which is ensured by their bound on the derivative (Theorem 1.19-(iii)). Thus
I
Hn (Xt,s (A)) ≥ lim sup Hn (Xt,s
(AI )).
I→∞
I (A )) in the sum over i ∈ I of Hn (X I (x )). Using the one to one
Let us decompose Hn (Xt,s
I
t,s i
correspondence of Lemma 2.3
ξp (τ )
= Hpp (t, xi , p) + O(t − τ )
τ −t
and
Therefore
and
ξp (τ + δ)
= Hpp (t, xi , p) + O(t − τ ).
τ +δ−t
ξp (τ ) ξp (τ + δ) τ − t − τ + δ − t ≤ O(t − τ ),
(
)
t − (τ + δ)
−1
ξp (τ )(ξp (τ + δ)) − Id ≤ O(t − τ ).
t−τ
Thus, passing to the determinant,
( )n (
)
1
t − (τ + δ) n
det(ξp (τ + δ)) ≥
det(ξp (τ )).
2
t−τ
47
SBV Regularity for Hamilton-Jacobi equations
From which it follows
H
n
I
(Xt,τ
+δ (xi ))
( )n (
)
1
t − (τ + δ) n n I
≥
H (Xt,τ (xi )).
2
t−τ
Summing up all the terms
H
n
I
(Xt,τ
+δ (AI ))
( )n (
)
1
t − (τ + δ) n n I
≥
H (Xt,τ (AI )).
2
t−τ
I (A )) = Hn (X (A)) and the Hausdorff convergence we
Finally using the fact that Hn (Xt,τ
t,τ
I
obtain
I
Hn (Xt,τ +δ (A)) ≥ lim sup Hn (Xt,τ
+δ (AI ))
I→∞
( )n (
)
1
t − (τ + δ) n n I
≥ lim sup
H (Xt,τ (AI ))
2
t−τ
I→∞
)
( )n (
t − (τ + δ) n n
1
=
H (Xt,τ (A)).
2
t−τ
Hence the thesis is proved.
2.2.5
Conclusion of the proof
The previous lemmas allow us to prove the following one.
We will denote the Cantor part of Dx2 u(t, ·) with Dc2 u(t, ·).
Lemma 2.21. For ε small enough, for any t in (τ, τ + ε] such that |Dc2 u(t, ·)|(Ωt ) > 0 and δ in
(0, τ + ε − t], there exists a Borel set A ⊂ Ωt such that
i) Hn (A) = 0, |Dc2 u(t, ·)|(A) > 0 and |Dc2 u(t, ·)|(Ωt \ A) = 0;
ii) Xt,τ is single-valued on A;
iii) and
χt,τ (A) ∩ χt+δ,τ (Ωt+δ ) = ∅.
Proof. From Proposition 1.15 and the definition of Cantor part of a measure, there exists a Borel
set A such that
• Dx+ u(t, x) is single-valued for every x ∈ A,
• Hn (A) = 0,
• |Dc2 u(t, ·)|(Ωt \ A) = 0 and |Dc2 u(t, ·)|(A) > 0.
48
2.2 Proof of Theorem 2.1
By contradiction suppose there exists a compact set K ⊂ A such that
|Dc2 u(t, ·)|(K) > 0
and
Xt,τ (K) = χt,τ (K) ⊂ χt+δ,τ (Ωt+δ ).
Call ω := |Dc2 u(t, ·)|(K).
Then there exists a Borel set K̃ ⊂ Ωt+δ such that χt,τ (K) = χt+δ,τ (K̃). Moreover, thanks
to the fact that we are considering classical characteristics starting from K̃, we have
χt+δ,t (K̃) = K
and
χt+δ,s (K̃) = χt,s (K) ∀s ∈ [τ, t).
Using Lemma 2.19, for any s ∈ [τ, t),
( )n (
)n
1
δ
H (K) = H (Xt+δ,t (K̃)) ≥
Hn (Xt+δ,s (K̃))
2
t+δ−s
( )n (
)n
1
δ
=
Hn (Xt,s (K)).
2
t+δ−s
n
Hence
n
)n
( )n (
δ
1
Hn (Xt,s (K)).
H (K) ≥
2
t+δ−s
n
(2.16)
Moreover if we choose s such that t − s is small enough
∫
Hn (Xt,s (K)) ≥ C1 Hn (K) − C2 (t − s)
d∆s u(t, ·) + O((t − s)2 )
K
∫
≥ − C2 (t − s)
d∆c u(t, ·) + O((t − s)2 )
K
≥ + C2 (t − s)ω + O((t − s)2 )
C2 2
ω ,
≥
2
where we have used the fact that Hn (K) = 0, that ∆j u(t, K) ≤ 0, which is true due to semiconcavity, implies ∆s u(t, K) ≤ ∆c u(t, K), and −∆c u(t, K) ≥ |Dc2 u(t, ·)(K)| = ω. Thus
Hn (Xt,s (K)) ≥
C2 2
ω .
2
Combining (2.16) with (2.17) we obtain
( )n (
)n
1
δ
C2 2
n
H (K) ≥
ω > 0.
2
t+δ−s
2
This is in contradiction with our hypothesis.
We now have all the necessary Lemmas to prove the Theorem 2.1.
49
(2.17)
SBV Regularity for Hamilton-Jacobi equations
Proof. For ε > 0 sufficiently small such that Lemmas 2.15, 2.18, 2.19, and 2.21 hold, consider
the functional F defined in (2.8) over the interval [τ, τ + ε]. F is bounded, and, from Lemma
2.15, F is a monotone function. Thus its points of discontinuity are at most countable.
We will prove that the presence of a Cantor part at a time t is related to a discontinuity of
the functional F in t, hence there must be only a countable number of t’s in [τ, τ + ε] for which
there is a Cantor part.
Suppose there exists a t in (τ, τ + ε) such that
|Dc2 u(t, Ωt )| > 0,
then for any δ > 0 let A be the set of Lemma 2.21. Using Lemma 2.21-(iii) we get
F (t + δ) ≤ F (t) − Hn (Xt,τ (A))
(2.18)
To compute Hn (Xt,τ (A)) call ω := |Dc2 u(t, ·)|(A). As we saw in the previous lemma, if we choose
s ∈ [τ, t) such that t − s is small enough, we have
Hn (Xt,s (A)) ≥
C2 2
ω .
2
Moreover for Lemma 2.19
( )n (
)
1
t−τ n n
H (Xt,τ (A)) ≥
H (Xt,s (A)).
2
t−s
n
Hence
( )n (
)
1
t − τ n C2 2
H (Xt,τ (A)) ≥
ω ≥ Cω 2 .
2
t−s
2
n
We can now use this estimate in (2.18) obtaining
F (t + δ) ≤ F (t) − Cω 2 .
Letting δ → 0
lim sup F (t + δ) < F (t).
δ→0
Therefore t is a point of discontinuity for F , as needed.
As already noticed this concludes the proof of our theorem, since F can have only a countable
number of points of discontinuity.
50
Chapter 3
Jacobian’s regularity
Let us consider a viscosity solution u of the Hamilton-Jacobi equation
∂t u + H(Dx u) = 0
with bounded Lipschitz initial datum u0 (x) and uniformly convex Hamiltonian H.
Taken a Borel set B ⊂ Ωt we call Jacobian the measure J(t, ·) defined as
J(t, B) := Hn (Dx+ u(t, B)).
Since Dx u(t, ·) is SBV out of a countable number of t’s, the Jacobian J cannot have a
positive part between Hn and Hn−1 , i.e., out of a countable number of t’s, the measure J(t, ·)
has only an absolute continuous part with respect to Hn and a part which is concentrated on a
Hn−1 -rectifiable set and is absolute continuous with respect to Hn−1 .
One can wonder if the Jacobian has only integer parts, that is, out of a countable number of
t’s, J(t, ·) can have only parts which are concentrated on a Hk -rectifiable set and are absolute
continuous with respect to Hk for k ∈ {0, 1, . . . , n}.
The following counterexample shows that this cannot be true.
Example 3.1. Let V : R → [0, 1] be the Vitali function, and consider the concave function
defined for s ∈ [0, +∞), y ∈ R, α, β positive constants
∫ y
v(s, y) = −α
V (z)dz − βs.
0
Note that the y-derivative of this function is precisely the Vitali function, hence a function which
log 2
sends the Cantor set C ⊂ [0, 1], a set of positive H log 3 -measure, in a set of positive H1 -measure,
precisely H1 (V (C)) = 1. Moreover its derivative is a measure which gives a positive value to
the Cantor set, V ′ (C) = 1.
We construct a viscosity solution of the two-dimensional Hamilton-Jacobi equation
1
∂t u + |Dx u|2 = 0,
2
(3.1)
whose behavior on vertical sections is exactly the behavior of the function v, i.e. for any fixed
x1 u(·, (x1 , ·)) = v(·, ·).
51
Jacobian’s regularity
Note that in this case the function
Xt,0 (x) := x − tHp (Dx+ u(t, x))
defined on Ωt is precisely
Xt,0 (x) = x − tDx+ u(t, x).
(
)
x−Xt,0 (x)
Hence J(t, x) = Hn
.
t
Let us construct the initial datum for our viscosity solution as
{
}
y12 + (y − y2 )2
u(0, (y1 , y2 )) :=
max
v(s, y) −
.
2s
s∈[0,+∞),y∈R
We are looking for an explicit form of this function so that it can be easily taken as the initial
datum of our Hamilton-Jacobi equation.
To find a maximizer we take the derivatives with respect to s and y and equal them to 0,
finding
y 2 + (y − y2 )2
y − y2
= 0,
−β + 1
= 0, −αV (y) −
2s2
s
from which we deduce that (s, y) can be a maximizer for u(0, (y1 , y2 )) if these relations are
invertible
√
y2 = y + αsV (y), y1 = ±s 2β − α2 V (y)2 .
(3.2)
2
This is possible if β ≥ α2 . In this case the above system admits a solution, i.e. for every
(y1 , y2 ) one can find (s, y) that solve the system.
Let us verify that s, y are a true maximizer for our function
F (s, y) := v(s, y) −
y12 + (y − y2 )2
.
2s
The Jacobian matrix of F (s, y) evaluated at s, y is
] [
[ 2
y1 +(y−y2 )2
y−y2
− 2β
− αVs(y)
−
s
s3
s2
=
y−y2
−αV ′ (y) − 1s
− αVs(y) −αV ′ (s) −
s2
]
1
s
.
2
For β ≥ α2 , it has exactly two negative eigenvalues, thus (s, y) as in the system (3.2) is the
unique global maximizer for u(0, (y1 , y2 )).
For simplicity we set β = 1, α = 1.
With this parametrization the initial datum takes the form
∫ y
))
( ( √
2
=−
V (z)dz − 2s.
u 0, ±s 2 − V (y) , y + sV (y)
0
We can now recover the unique viscosity solution of the Hamilton-Jacobi equation with this
initial datum through the Hopf-Lax formula
}
{
(x1 − y1 )2 + (x2 − y2 )2
u(t, (x1 , x2 )) = min
,
(3.3)
u(0, (y1 , y2 )) +
2t
(y1 ,y2 )∈R2
52
which, for x1 ≥ 0, is equivalent to
u(t, (x1 , x2 )) =
min
s∈[0,+∞),y∈R
{ ∫
−
y
V (z)dz − 2s
√
}
(x1 − s 2 − V (y)2 )2 + (x2 − y + sV (y))2
+
.
2t
0
To find the minimizer and obtain an explicit formula for our solution, take the derivatives
in s and y and let them be equal to zero. What we obtain is
[
]
√
1 √
−2 + (s 2 − V (y)2 − x1 ) 2 − V (y)2 + (y + sV (y) − x2 )V (y) = 0
t
and
[
]
√
1
V (y)V ′ (y)
′
2
− (s 2 − V (y) − x1 ) √
−V (y) +
+ (y + sV (y) − x2 )(1 + sV (y)) = 0,
t
2 − V (y)2
where V ′ (y) has to be intended as the distributional derivative of V which is null for H1 -a.e. y
in R.
Therefore taken s and y, with V ′ (y) = 0, they are a minimizer for u(t, (x1 , x2 )) where
√
x2 = y + (s − t)V (y),
x1 = ±(s − t) 2 − V (y)2 ,
and
∫
( (
))
√
2
u t, ±(s − t) 2 − V (y) , y + (s − t)V (y) = −
y
V (z)dz − 2(s − t).
0
The derivative of such a solution is SBV and its Jacobian J(t, ·) has no positive part in the
interval (1, 2) out of a countable number of t’s.
Let us see that J(t, ·) has a positive part between H1 and H0 for every t > 0.
Taking x1 = 0 and x2 = y in the Cantor set C, i.e. (x1 , x2 ) ∈ E := {0} × C, we compute the
minimizers for (t, (0, y)) in (3.3). From the previous computation
∫ y
u(t, (0, y)) = −
V (z)dz.
0
(
)
√
Hence the two minimizers are y1 = ±t 2 − V (y)2 , y2 = y + tV (y) .
Thus for every t > 0 and every y ∈ C
√
√
Dx+ u(t, (0, y)) = [− 2 − V (y)2 , 2 − V (y)2 ] × {V (y)}.
log2
Therefore, we have found a set E which is of positive H log3 -measure such that J(t, x) = 0
for every x ∈ E and every t > 0. Moreover, call
}
{ √
√
2
2
A := [− 2 − V (y) , 2 − V (y) ] × {V (y)}| y ∈ C ⊂ R2 ,
then A = Dx+ u((0, E) and J(t, E) = H2 (A) > 0.
53
Jacobian’s regularity
Hence the Jacobian has a positive part between H1 and H0 .
Observe also that the set
√
√
[−t 2 − V (y)2 , t 2 − V (y)2 ] × {y + tV (y)}
does not contain the set
[−t′
√
2 − V (y)2 , t′
√
2 − V (y)2 ] × {y + t′ V (y)}
for t′ > t.
54
Chapter 4
SBV-like regularity for
Hamilton-Jacobi equations: the
convex case
In this chapter we consider the Hamilton-Jacobi equation
∂t u + H(Dx u) = 0
in Ω ⊂ [0, T ] × Rn ,
where H is a smooth convex Hamiltonian. A viscosity solution of such an equation is locally
Lipschitz but in general it doesn’t have any additional regularity. As already seen in the uniformly convex case instead the viscosity solution u is semiconcave, therefore Dx u(t, ·) belongs
to BV and Dx2 u(t, ·) is a matrix of Radon measures. Moreover, in [9], Bianchini, De Lellis and
Robyr proved the theorem here presented as Theorem 0.2, which states that Dx u(t, ·) belongs
to [SBV (Ωt )]n , Ωt := {x ∈ Rn | (t, x) ∈ Ω}, out of a countable number of t’s in [0, T ]. More
precisely Dx2 u(t, ·) can have Cantor part only for a countable number of t’s in [0, T ].
When H is just convex, Dx u(t, ·) looses its BV regularity, an example can be found in Remark
3.7 in Bianchini [8]. However, in this chapter, we show that an SBV-like regularity result can
be proven for the vector field
d(t, x) := Hp (Dx u(t, x)),
defined on the set U of points (t, x) where u(t, x) is differentiable in x. Here Hp is the gradient
of the Hamiltonian H(p). Indeed the divergence divd(t, ·) is in general a locally finite Radon
measure. When the vector field d(t, ·) is BV and suitable hypotheses are made on the Lagrangian
L, the Legendre transform of H, the measure divd(t, ·) has Cantor part only for a countable
number of t’s in [0, T ].
More precisely let H be C 2 (Rn ), convex and such that lim|p|→∞ H(p)
|p| = +∞.
(HYP(0)) Suppose the vector field d(t, ·) belongs to [BV (Ωt )]n for every t ∈ [0, T ].
Define Vπn as
Vπn := {v ∈ Rn | L(·) is not twice differentiable in v},
and
Σπn := {(t, x) ∈ U | d(t, x) ∈ Vπn }
55
and Σcπn := U \ Σπn .
SBV-like regularity for Hamilton-Jacobi equations: the convex case
(HYP(n)) We suppose Vπn to be contained in a finite union of hyperplanes Ππn .
For j = n, . . . , 3 for every (j − 1)-dimensional plane πj−1 in Ππj , let Lπj−1 : Rj−1 → R be
the (j − 1)-dimensional restriction of L to πj−1 and
Vπj−1 := {v ∈ Rj−1 | Lπj−1 (·) is not twice differentiable in v}.
Define
Σπj−1 := {(t, x) ∈ Σπj | d(t, x) ∈ Vπj } and
Σcπj−1 := Σπj \ Σπj−1 .
(HYP(j-1)) We suppose Vπj−1 is contained in a finite union of (j − 2)-dimensional planes Ππj−1 ,
for every πj−1 ∈ Ππj .
Theorem 4.1. Under the above assumptions (HYP(0)),(HYP(n)),...,(HYP(2)), the Radon
measure divd(t, ·) has Cantor part on Ωt only for a countable number of t’s in [0, T ].
This result can be seen as the multi-dimensional version of Theorem 0.4 proved by Robyr (see
[35] for its proof). Furthermore, we prove that in the one-dimensional case the BV regularity of
d(t, x), which was an hypothesis in the theorem of Robyr, follows automatically in the case of a
convex smooth Hamiltonian.
The question on the SBV regularity of d(t, ·) without any additional hypothesis is still open.
The chapter is organized as follows. In Section 4.1 we extend the definition of the vector field
d to the all Ω, we prove that divd(t, ·) is a locally finite Radon measure on Ωt , for all t ∈ [0, T ].
In Section 4.2 we present the general strategy used to prove that divd(t, ·) has a Cantor part only
for a countable number of t′ s in [0, T ]. In Section 4.3 we study the one-dimensional case and we
prove that divd(t, ·) belongs to SBV (Ωt ), out of a countable number of t’s in [0, T ], without any
additional hypothesis. In Section 4.4 we study the multi-dimensional case and prove Theorem
4.1. We also state some easy corollaries.
4.1
Extension and preliminary properties of the vector field d
We consider a viscosity solution u of the Hamilton-Jacobi equation
∂t u + H(Dx u) = 0 in Ω ⊂ [0, T ] × Rn ,
where H is C 2 (Rn ) convex and
lim
|p|→∞
H(p)
= +∞.
|p|
As already noticed, thanks to the time invariance of the equation and to Proposition 1.18,
it is enough to consider the unique viscosity solution of the following Cauchy problem
{
∂t u + H(Dx u) = 0 in Ω ⊂ [0, T ] × Rn ,
u(0, x) = u0 (x) for all x ∈ Ω0 ,
where u0 (x) is a bounded Lipschitz function on Ω0 .
The vector field d(t, x) = Hp (Dx u(t, x)) is well defined where u(t, x) is differentiable in x,
i.e. Hn -a.e. on Ωt , for every t ∈ [0, T ].
56
4.1 Extension and preliminary properties of the vector field d
Thanks to the Lipschitz regularity of u(t, ·) and the fact that H is smooth, the vector field
d(t, ·) belongs to [L∞ (Ωt )]n .
Moreover d is constant along optimal rays. Indeed, thanks to Theorem 1.31-(iii), we have
d(t, x) = d(s, x − (t − s)d(t, x))
for all 0 ≤ s ≤ t.
A natural extension of d to Ω is D(·) : Ω → Rn
{
}
x−y
+
D(t, x) :=
| y is a minimum for ut,0 (t, z) ,
t
where u+
t,0 is the forward solution as in Definition 1.42.
D(t, x) is a multi-valued function which coincides with d(t, x) in the points (t, x) where u(t, x)
is differentiable in x. Indeed, where u(t, ·) is differentiable, u(t, x) = u+
t,0 (t, x) and they both
admit as unique minimizer y = x − tHp (Dx u(t, x)) in Ω0 .
Following the results of Bianchini and Gloyer in [10], we can prove that D(t, x) has closed
graph and thanks to the fact that D(t, x) is closed
D(t, x) ⊂ D(t, x′ ) + B(0, ε)
for x, x′ ∈ Ωt . Moreover D(t, x) is a Borel measurable function and divd(t, ·) a locally finite
Radon measure. We repeat the proof for the reader’s convenience.
Theorem 4.2. For every t ∈ (0, T ], the divergence divd(t, ·) is a locally finite Radon measure
with negative singular part.
Proof. Consider an approximation of our vector field done by taking a dense sequence of points
I
{yi }∞
i=1 in Ω0 . Fix an integer I > 0, call Ω0 := {yi | i = 1, . . . , I} and define for any x ∈ Ωt
{
(
)}
x − yi
−
u+
(t,
x)
:=
min
u
(0,
y
)
+
tL
,
i
t,0
I
i∈I
t
where u−
t,0 is the backward solution as in Definition 1.42.
Through this approximation the set Ωt is split into at most I open regions Ωit , i = 1, . . . , I,
defined by
Ωit := interior of {x ∈ Ωt | ∃yi minimizer for u+
I (t, x)},
together with the set
JtI :=
)
∪( i
j
Ωt ∩ Ωt
i̸=j
of negligible Hn -measure. Indeed, even for u+
I (t, ·) the set of points with more than one minimum
n
is the set of points of non differentiability of u+
I (t, ·) and this set has H -measure zero. We define
the vector field dI on Ω so that on each open set Ωit
dI (t, x) :=
57
x − yi
.
t
SBV-like regularity for Hamilton-Jacobi equations: the convex case
Using explicitly the definition of dI and the fact that Hn (JtI ) = 0,
divdI (t, x) ≤
n
.
t
Thanks to the pointwise convergence of dI to d
divd(t, ·) −
n n
H ≤ 0,
t
i.e. divd(t, ·) − nt Hn is a negative definite distribution, hence it is a locally finite Radon measure.
Thus divd(t, ·) is itself a locally finite Radon measure.
Moreover
n
divd(t, ·) ≤ Hn ,
t
implies that the singular part of this measure can be only negative.
From now on we will denote µ(t, ·) := divd(t, ·).
Since we have proven that µ(t, ·) is a locally finite Radon measure, it makes sense to ask
whether is possible or not that µ(t, ·) has Cantor part for all t in [0, T ]. Note that if a Cantor
part is different from zero then it must be negative for Theorem 4.2.
4.2
General strategy
In order to prove that µ(t, ·) has Cantor part only for a countable number of t’s, the general
idea is now standard, see [4], [9] and Chapter 2.
We present this strategy in the form useful in our case. We reduce to a smaller interval
[τ, T ], for a fixed τ > 0, and we construct, on this interval, a monotone bounded functional F (t).
Then, we relate the presence of a Cantor part for the measure µ(t, ·), for a certain t in [τ, T ],
with a jump of the functional F in t. Since this functional is bounded monotone it can have
only a countable number of jumps. Thus, the Cantor part of µ(t, ·) can be different from zero
only for a countable number of t’s.
To define F we consider the following maps: Xt,τ (x) : Ωt → Ωτ
Xt,τ (x) := x − (t − τ )D(t, x),
and its restriction to the set Ut of points where D(t, x) is single-valued, χt,τ (x) : Ut → Uτ
χt,τ (x) := x − (t − τ )d(t, x).
We will sometimes write χt,τ (Ωt ) for χt,τ (Ut ).
We define the functional F : (τ, T ] → R
F (t) := Hn (χt,τ (Ut )).
The functional F is bounded, and, due to the fact that optimal rays do not intersect except
than at time t or 0, F is a monotone decreasing functional.
In order to apply the strategy above we need two estimates of the following type:
58
4.2 General strategy
i) For any Borel set A ⊂ Ut for t in (τ, T ]
Hn (Xt,τ (A)) ≥ C1 Hn (A) − (t − τ )C2 µ(t, A),
(4.1)
where C1 , C2 are fixed positive constants.
ii) For any Borel set A ⊂ Ωt , for t in (τ, T ] and for every 0 ≤ δ ≤ t − τ
)
(
t − (τ + δ) m n
n
H (Xt,τ (A)),
H (Xt,τ +δ (A)) ≥
t−τ
(4.2)
where m ∈ N, m > 0 is fixed.
Indeed with the estimates above we can prove the following lemma.
Lemma 4.3. For any t in (τ, T ] such that µc (t, Ωt ) < 0 and δ in (0, T − t], there exists a Borel
set A ⊂ Ut such that
i) Hn (A) = 0, µc (t, A) < 0 and µc (t, Ωt \ A) = 0;
ii) Xt,τ is single-valued on A;
iii) and
χt,τ (A) ∩ χt+δ,τ (Ωt+δ ) = ∅.
Proof. The set of points where d(t, ·) is not single-valued, which coincides with the set of points
where u(t, ·) is differentiable, is an Hn−1 -rectifiable set, due to the Lipschitz regularity of u(t, ·).
Hence, the Radon measure µ(t, ·) has null Cantor part on it. This and the definition of Cantor
part of a measure imply the existence of a Borel set A such that
• d(t, x) is single-valued for every x ∈ A,
• Hn (A) = 0,
• µc (Ωt \ A) = 0 and µc (A) < 0.
By contradiction suppose there exists a compact set K ⊂ A such that
µc (t, K) < 0
and
Xt,τ (K) = χt,τ (K) ⊂ χt+δ,τ (Ωt+δ ).
Then there exists a Borel set K̃ ⊂ Ωt+δ such that χt,τ (K) = χt+δ,τ (K̃). Moreover, thanks
to the fact that we are considering optimal rays starting from K̃, we have
χt+δ,t (K̃) = K
and χt+δ,τ (K̃) = χt,τ (K).
Using the estimate (4.2),
(
H (K) = H (Xt+δ,t (K̃)) ≥
n
n
δ
t+δ−τ
)m
(
H (Xt+δ,τ (K̃)) =
n
59
δ
t+δ−τ
)m
Hn (Xt,τ (K)).
SBV-like regularity for Hamilton-Jacobi equations: the convex case
(
Hence
H (K) ≥
n
δ
t+δ−τ
)m
Hn (Xt,τ (K)).
Moreover applying estimate (4.1)
(
)m
δ
n
H (K) ≥
(C1 Hn (K) − (t − τ )C2 µ(t, A)) .
t+δ−τ
Since Hn (K) = 0 we obtain µ(t, A) ≥ 0 in contrast with the fact that µc (t, A) < 0.
The estimate (4.1) and Lemma 4.3 lead us to the expected conclusion.
Suppose there exists a t in (τ, T ) such that
µc (t, Ωt ) < 0,
then, for any δ > 0, let A be the set of Lemma 4.3. According to Lemma 4.3-(iii) we have
F (t + δ) ≤ F (t) − Hn (Xt,τ (A)).
Moreover, the estimate (4.1) gives
F (t + δ) ≤ F (t) + (t − τ )C2 µc (t, A).
Hence, letting δ → 0, we obtain
lim sup F (t + δ) < F (t).
δ→0
Therefore t is a point of discontinuity for F , as we wanted to prove.
4.3
One-dimensional case
We first consider the one-dimensional case. In this case we don’t need any further assumption
on d or L to prove the following theorem.
Theorem 4.4. The vector field d(t, ·) belongs to SBV (Ωt ), out of a countable number of t ∈
[0, T ].
In the uniformly convex case, Theorem 4.4 is a corollary of Theorem 0.1 of the Introduction
proved by Ambrosio and De Lellis in [4].
∂
Proof. Since we are in the one-dimensional case, divd(t, x) = ∂x
d(t, x). Hence, Theorem 4.2
implies that d(t, x) belongs to BV (Ωt ), for every t ∈ (0, T ].
Moreover, D(t, ·) is semimonotone. Indeed, since we are following optimal rays for u+
t,0 , they
do not intersect except than at time 0 or t. Thus for x1 , x2 ∈ Ωt , x1 < x2 and d1 ∈ D(t, x1 ), d2 ∈
D(t, x2 ), it must hold
x1 − td1 ≤ x2 − td2 ,
60
4.3 One-dimensional case
otherwise the rays cross each other at a time s ∈ (0, t). Hence the function 1t x − D(t, x) is
monotone increasing and D(t, x) is semimonotone with constant C = 1t .
Let us consider the map Xt,τ for any t ∈ (τ, T ], τ > 0 fixed. The fact that we are in the
one-dimensional case implies that for t, x, such that D(t, x) is multi-valued,
D(t, x) = [d1 , d2 ],
¯ d˜ ∈ [d1 , d2 ],
where d1 , d2 ∈ R are the speeds of the optimal rays for u(t, x). Indeed, for every d,
¯
˜
the ray [x, x − td] cannot cross [x, x − td], since they are straight lines starting in the same point.
So they fill the triangle delimited by [x, x − td1 ], [x, x − td2 ]. Moreover, optimal rays starting
in other points cannot cross [x, x − td1 ] and [x, x − td2 ], at intermediate time, since they are
optimal. Thus they cannot cross any other ray [x, x − td], where d ∈ [d1 , d2 ]. For this reason
these rays are optimal for u+
t,0 (t, x). Thus optimal rays for the forward solution completely fill
the set {Ωs | s ∈ [0, t]}.
Remark 4.5. This argument holds also in the multi-dimensional case but only for a set of
points of non differentiability of zero-dimension. The argument is not true in general when the
points of non differentiability lie on a surface of dimension greater than zero, since rays starting
in two different points of this surface can intersect even at intermediate times.
The above consideration ensures that the map Xt,τ is injective for τ > 0, however this map
is multi-valued. To recover the Lipschitzianity we use the Hille-Yosida transformation as seen
in [1] and Chapter 2.
For any Borel set A ⊂ Ωt , let z ∈ B := A + T (A), T (x) := (Cx − D(t, x)) and w(z) :=
(Id1 + (T )−1 )−1 (z). Then the following 1-Lipschitz transformations
{
x(z) = z − w(z)
(4.3)
p(z) = Cz − (C + 1)w(z),
transform our graph
{(x, p)| x ∈ A, p ∈ D(t, x)}
into the equivalent graph of a maximal monotone function
{(z − w(z), Cz − (C + 1)w(z))| z ∈ B}.
Recall that C is the semimonotonicity constant of D(t, ·).
Following optimal rays starting in A with speed in D(t, A), we can now pass from Xt,τ (x) to
a Lipschitz map defined on B
ξ(τ, z) := z − w(z) − (t − τ )(Cz − (C + 1)w(z)).
Note that
{(Cz − (C + 1)w(z))| z ∈ x + T (x)} = D(t, x)
so that Xt,τ (x) = {ξ(τ, z)| z ∈ x + T (x)} and Xt,τ (A) = ξ(τ, B).
We can now apply the Area Formula to ξ(τ, ·)
∫
∫
0
−1
H (ξ(τ, ·) (w))dw =
|ξz (τ, z)|dz.
ξ(τ,B)
B
61
(4.4)
SBV-like regularity for Hamilton-Jacobi equations: the convex case
Thanks to the injectivity of the map Xt,τ , which is preserved when passing to the Lipschitz
parametrization, the left term of (4.4) is precisely the measure of the set ξ(τ, B). Hence, we
have
∫
H0 (ξ(τ, ·)−1 (w))dw = H1 (ξ(τ, B)) = H1 (Xt,τ (A)).
ξ(τ,B)
Moreover, differentiating ξ we respect to z we denote
ξz (τ, z) = ξz (t, z) − (t − τ )ξ˙z (t, z),
∂
∂
where ξz (t, z) := ∂z
(z − w(z)) and ξ˙z (t, z) := ∂z
(Cz − (C + 1)w(z)).
Thus we have
∫
∫
∫
1
˙
H (Xt,τ (A)) =
|ξz (t, z) − (t − τ )ξz (t, z)|dz ≥
ξz (t, z)dz − (t − τ )
ξ˙z (t, z)dz.
B
Observing that
B
∫
∫
ξ˙z (t, z)dz =
B
B
B
∂
(Cz − (C + 1)w(z))dz = µ(t, A),
∂z
we have proven the following estimate: given a Borel set A ⊂ Ωt for t in (τ, T ], we have
H1 (Xt,τ (A)) ≥ H1 (A) − (t − τ )µ(t, A).
(4.5)
Moreover, since for every 0 ≤ δ ≤ t − τ
ξz (t, z) − (t − (τ + δ))ξ˙z (t, z) =
δ
t − (τ + δ)
ξz (t, z) +
(ξz (t, z) − (t − τ )ξ˙z (t, z)),
t−τ
t−τ
and ξz (t, z) > 0, we have
t − (τ + δ)
ξz (t, z) − (t − (τ + δ))ξ˙z (t, z) ≥
(ξz (t, z) − (t − τ )ξ˙z (t, z)).
t−τ
Thus, integrating the last equation over B, we obtain the following estimate: given a Borel set
A ⊂ Ωt for t in (τ, T ], then for every 0 ≤ δ ≤ t − τ we have
H1 (Xt,τ +δ (A)) ≥
t − (τ + δ) 1
H (Xt,τ (A)).
t−τ
(4.6)
The estimates (4.5) and (4.6) are of type (4.1) and (4.2) respectively, thus they are enough
to prove the SBV regularity of d, as seen in Subsection 4.2.
4.4
The multi-dimensional case
In [9] Bianchini, De Lellis and Robyr proved that the estimates (4.1) and (4.2) hold for the
uniformly convex Hamiltonian Hϵ (p) := H(p) + 2ϵ |p|2 for every ε > 0 in a small interval of time
and with constants strictly depending on ϵ. Thus, the two estimates cannot pass to the limit.
62
4.4 The multi-dimensional case
Nevertheless, we can prove that the divergence divd(t, ·) has Cantor part only for a countable
number of t’s, adding some hypothesis on the regularity of d and on the structure of the the set
of points where L is not twice differentiable.
As already noticed, the Lagrangian corresponding to a smooth convex Hamiltonian is strictly
convex but non smooth in general. Particular conditions on the set of points where L is not twice
differentiable will allow us to reduce iteratively our problem to a problem of lower dimension,
down to the one-dimensional case, where, as we have seen, SBV regularity can be proven without
additional assumptions.
Before going on with the proof we set some notations. We will denote with (x1 , x2 , . . . , xn )
the components of the vector x ∈ Rn and, to contract the notation, for a fixed j = 1, . . . , n − 1
we call x̂ ∈ Rn−j the vector defined so that
(x1 , . . . , xj , x̂) = (x1 , x2 , . . . , xn ).
Given a set E ⊂ [0, T ] × Rn we will denote with
Et := {x ∈ Rn | (t, x) ∈ E}
and for j = 1, . . . , n − 1
{
}
Ex1 ,...,xj := (t, xj+1 , . . . , xn )| (t, x1 , . . . , xj , xj+1 , . . . xn ) ∈ E .
As before we will sometimes denote with µ(t, ·) the Radon measure divd(t, ·) defined on Ωt .
(HYP(0)) Suppose that the vector field d(t, ·) belongs to [BV (Ωt )]n for any t ∈ [0, T ].
The measure divd can have Cantor part only on a subset of the points of differentiability in
x of u(t, x), i.e. the points where D(t, x) is single-valued. Thus we can reduce to the study of
our measure on the set
U := Ω \ {(t, x)| D(t, x) is multi-valued}.
Call V the set of points where L is not twice differentiable:
V := {v ∈ Rn | L(·) is not twice differentiable in v}.
Then the set U can be split into two subsets:
Σ := {(t, x) ∈ U | d(t, x) ∈ V }
and Σc := U \ Σ.
(HYP(n)) Suppose V is contained in a finite union of hyperplanes.
Claim 1.(n) The vector field d(t, ·) belongs to [SBV (Σct )]n out of a countable number of t’s
in [0, T ].
Claim 2.(n) The Radon measure divd(t, ·), restricted to Σt , can have Cantor part only for
a countable number of t’s in [0, T ].
The regularity of divd will follow from the previous claims and the fact that U = Σ ∪ Σc .
63
SBV-like regularity for Hamilton-Jacobi equations: the convex case
Proof of Claim 1.(n). For a fixed (t̄, x̄) ∈ Σc , the Hessian of L exists and is continuous in
v̄ := d(t̄, x̄). Thus there exist r > 0 and a (n + 1)-dimensional ball Brn+1 (t̄, x̄) ⊂ Ω \ Σ where L
and H are uniformly convex.
We can also find an open cone Cn+1 (t̄, x̄) ⊂ Brn+1 (t̄, x̄), properly containing (t̄, x̄), over which
an Hamilton-Jacobi equation can be solved. Indeed, we take an n-dimensional ball as base,
B n ⊂ (Brn+1 (t̄, x̄))t̄−σ ⊂ (Ω \ Σ)t̄−σ ,
for a certain 0 < σ < r, and we fix the height of length l ∈ R, 0 < l < 2r. The height must be
chosen according to the speed of propagation of the solution and such that t̄ < t̄ − σ + l.
Consider now the viscosity solution ū of the Cauchy problem
{
∂t ū + H(Dx ū) = 0
in Cn+1 (t̄, x̄),
ū(t − σ, x) = u(t − σ, x)1B n (x),
where 1E (x) is the indicator function of the set E. Note that u(t, x) = ū(t, x) on Cn+1 (t̄, x̄).
Thanks to the uniform convexity of H over Cn+1 (t̄, x̄), the main theorem of [9] ensures that
the vector field
¯ ·) := Hp (Dx ū(t, ·))
d(t,
is SBV out of a countable number of t’s in [t̄ − σ, t̄ − σ + l].
¯ ·) are both BV and coincide on (Cn+1 (t̄, x̄))t , thus, for
The vector fields d(t, ·) and d(t,
Proposition 1.9,
¯ x).
Dx d(t, ·) = Dx d(t,
Therefore d(t, ·) belongs to SBV ((Cn+1 (t̄, x̄))t ) out of a countable number of t’s in [t̄−σ, t̄−σ+l].
Finally, using the fact that Rn is a countable union of bounded sets, we can apply Besicovitch
covering Theorem, see [5], to prove that the set Σc can be fully covered by a countable number
i
, for i ∈ N, with the property stated above. Thus d(t, ·) belongs to [SBV (Σct )]n
of cones Cn+1
out of a countable number of t’s in [0, T ].
We consider now the behavior of divd on the set Σ. In order to prove Claim 2.(n), in the
n-dimensional case, n > 2, we need some other hypothesis on L and its restriction to the set
of points where L is not twice differentiable. No additional hypotheses are needed in the case
n = 2.
Proof of Claim 2.(n). 2-dimensional case. First, suppose V is a single straight line. Without
loss of generality we can fix V = {v ∈ R2 | v1 = 0}.
Call LV : R → R the restriction of the Lagrangian L to V ,
LV (v2 ) := L(0, v2 )
for any v2 ∈ R. Call I ⊂ R the set of every x1 in R such that Σx1 is non empty. Note that if
(t, x2 ) ∈ Σx1 then (0, x2 −td2 (t, (x1 , x2 ))) belongs to Σx1 because d(t, (x1 , x2 )) = (0, d2 (t, (x1 , x2 ))).
For every x1 ∈ I, we consider the one-dimensional Hamilton-Jacobi equation for the function
ux1 (t, x2 ).
{
∂t ux1 + HV (Dx2 ux1 ) = 0 in Σx1 ,
ux1 (0, x2 ) = u(0, (x1 , x2 )) ∀x2 ∈ (Σx1 )0 ,
64
4.4 The multi-dimensional case
where HV (p) is the Hamiltonian associated to LV (v).
The viscosity solution ux1 (t, x2 ) is equal to u(t, (x1 , x2 )) for every (t, (x1 , x2 )) ∈ Σ. Indeed
{
(
)}
x2 − y2
ux1 (t, x2 ) = min u(0, x1 , y2 ) + tLV
y2 ∈R
t
{
(
)}
x2 − y2
= min u(0, x1 , y2 ) + tL 0,
y2 ∈R
t
= u(t, (x1 , x2 )),
where the last equality follows from the fact that, for (t, x) in Σ, the unique minimizer in the
representation formula (1.10) is y = (x1 − td1 (t, x), x2 − td2 (t, x)) and d(t, x) = (0, d2 (t, x)).
Let us define as usual
dx1 (t, x2 ) := (HV )p2 (Dx2 ux1 (t, x2 ))
and
µx1 (t, ·) :=
∂
dx (t, ·).
∂x2 1
The vector field dx1 (t, ·) is one-dimensional. Hence, for Theorem 4.2, dx1 (t, ·) belongs to
BV ((Σx1 )t ) for any x1 ∈ I, for any t ∈ [0, T ].
On the set Σ ⊂ U , the matrix of Radon measures Dx d has no jump part. Moreover, since
Σt is contained on the set {x| d1 (t, x) = 0} and d(t, ·) is BV, Proposition 1.10 implies
∂
d1 (t, Σt ) = 0 and
∂x1
Therefore
divd(t, ·) =
∂
d1 (t, Σt ) = 0.
∂x2
∂
d2 (t, ·) on Σt .
∂x2
For every (t, x) ∈ Σ, ux1 (t, x2 ) = u(t, (x1 , x2 )) implies
d2 (t, x) = dx1 (t, x2 ).
The vector field d2 (t, (x1 , ·)) is a one-dimensional restriction of d2 (t, ·) thus, for Proposition 1.11,
belongs to BV ((Σx1 )t ) for H1 -a.e. x1 ∈ I. Since even dx1 (t, ·) is BV on (Σx1 )t , Proposition 1.9
implies
∂
∂
d2 (t, (x1 , ·)) =
dx (t, ·)
∂x2
∂x2 1
for H1 -a.e. x1 ∈ I. Therefore taken a Borel set A ⊂ Σt and any ϕ ∈ Cc∞ (Σt ),
∫
∫ ∫
ϕ(x)dµ(t, x) =
ϕ(x)dµx1 (t, x2 )dx1 .
A
I
Ax1
Thanks to the convexity of LV , we can apply Theorem 4.4 to µx1 (t, ·) and obtain the following
estimates.
For any τ > 0, let A be a Borel set in Σt , for t ∈ (τ, T ]. Then for any 0 ≤ δ ≤ t − τ , and
every section Ax1 , for x1 ∈ I, we have
65
SBV-like regularity for Hamilton-Jacobi equations: the convex case
x1
H1 (Xt,τ
(Ax1 )) ≥ H1 (Ax1 ) − (t − τ )µx1 (t, Ax1 ),
t − (τ + δ) 1 x1
H (Xt,τ (Ax1 )).
t−τ
x1
Here we denote with Xt,τ
(x2 ) the one-dimensional map defined on (Σx1 )t
x1
H1 (Xt,τ
+δ (Ax1 )) ≥
x1
Xt,τ
(x2 ) := x2 − (t − τ )dx1 (t, x2 ).
The corresponding 2-dimensional map
Xt,τ (x) := x − (t − τ )d(t, x),
reduces to
x1
Xt,τ (x) = (x1 , Xt,τ
(x2 ))
for every x ∈ Σt .
We can integrate the previous estimates with respect to H1 on I ⊂ R to recover estimates
of type (4.1) and (4.2).
For any τ > 0, given a Borel set A ⊂ Σt , for t in (τ, T ], we have
H2 (Xt,τ (A)) ≥ H2 (A) − (t − τ )µ(t, A).
(4.7)
For any τ > 0, given a Borel set A ⊂ Σt , for t in [τ, T ] and 0 ≤ δ ≤ t − τ we have
H2 (Xt,τ +δ (A)) ≥
t − (τ + δ) 2
H (Xt,τ (A)).
t−τ
(4.8)
Thus the strategy seen in the Subsection 4.2 can be easily applied to prove that µ(t, ·),
restricted to Σt , can have Cantor part only for a countable number of t’s in [0, T ].
Remark 4.6. Note that in this case nothing can be said about the Cantor part of
Thus we cannot say that d(t, ·) belongs to [SBV (Ωt )]2 .
∂
∂x1 d2 (t, ·).
Consider now the case in which V consists of a finite number of straight lines. When we
consider µ(·, ·) restricted to the points of Σ such that d(t, x) belongs only to a part of one of
the straight lines, we can apply the considerations done in the case where V consists only of
a single straight line. On the other hand, when we consider µ(·, ·) restricted to the points of
Σ such that d(t, x) belongs to an intersection point (v1 , v2 ) of two, or more, straight lines, the
divergence divd(t, ·) must be zero on every Borel subset of {x| d1 (t, x) = v1 , d2 (t, x) = v2 }, for
Proposition 1.10. Thus the measure µ(t, ·), restricted to Σt , can have Cantor part only for a
countable number of t’s in [0, T ] even when V consists of a finite number of straight lines. The
case in which V is contained in a finite number of straight lines is analogous.
n-dimensional case. We prove the claim iterating a subdivision of Σ down to the dimension
one.
Call Vn := V . At the step n − j, for j = n, . . . , 3, we first suppose that Vj consists of a single
(j − 1)-dimensional plane, without loss of generality we can fix
Vj = {v ∈ Rn | v1 = 0, . . . , vn+1−j = 0}.
66
4.4 The multi-dimensional case
Call LVj : Rj−1 → R the restriction of LVj+1 to Vj ,
LVj (v̂) := LVj+1 (0, v̂) = L(0, . . . , 0, v̂)
for any v̂ ∈ Rj−1 .
(HYP(j-1)) We require that the restriction LVj is twice (j − 1)-differentiable out of the set
Vj−1 ,
Vj−1 := {v̂ ∈ Rj−1 | LVj (·) is not twice differentiable in v̂},
and Vj−1 is contained in a finite number of (j − 2)-dimensional planes.
Then we can subdivide Σj into two set:
Σj−1 := {(t, x) ∈ Σj | d(t, x) ∈ Vj−1 }
and Σcj−1 := Σj \ Σj−1 .
Thus, at every step, we have to prove the following claims.
Claim 1.(j-1) The Radon measure divd(t, ·), restricted to (Σcj−1 )t , can have Cantor part
only for a countable number of t’s in [0, T ].
Claim 2.(j-1) The Radon measure divd(t, ·), restricted to (Σj−1 )t , can have Cantor part
only for a countable number of t’s in [0, T ].
Proof of Claim 1.(j-1) . We will prove it for j = n, in the other cases the proof is similar.
For a fixed (t̄, x̄) ∈ Σcn−1 , the Hessian of LV exists and is continuous in v̂ := (d2 (t̄, x̄), . . . , dn (t̄, x̄)) ∈
n−1
R
. Thus there exist r > 0 and a (n + 1)-dimensional ball Brn+1 (t̄, x̄) ⊂ Ω \ Σn−1 where LV
and HV are uniformly convex.
We can also find, as we did in the proof of Claim 1.(n), an open cone Cn+1 (t̄, x̄) ⊂ Brn+1 (t̄, x̄)
of height [t̄ − σ, t̄ − σ + l], for a certain 0 < σ < r, t̄ < t̄ − σ + l and base B n , which contains
properly (t̄, x̄). On every section (Cn+1 (t̄, x̄))x1 , for every x1 ∈ I := {z ∈ R| (Cn+1 (t̄, x̄))z ̸= ∅},
we can consider the viscosity solution ūx1 of the (n − 1)-dimensional Hamilton-Jacobi equation
{
∂t ūx1 + HV (Dx̂ ūx1 ) = 0 in (Cn+1 (t̄, x̄))x1 ,
ūx1 (t̄ − σ, x̂) = u(t̄ − σ, x)1B n (x).
As usual we define
d¯x1 (t, x̂) := (HV )p̂ (Dx̂ ūx1 (t, x̂)).
and
µ̄x1 (t, ·) := divn−1 d¯x1 (t, ·).
The vector field d¯x1 (t, ·) belongs to [BV (((Cn+1 (t̄, x̄))x1 )t )]n−1 for any x1 ∈ I, for any t ∈
[t̄ − σ, t̄ − σ + l]. Indeed in every (Cn+1 (t̄, x̄))x1 HV is uniformly convex.
Since we have a uniform convexity constant for HV , which holds on every (Cn+1 (t̄, x̄))x1 , for
x1 ∈ I, we can arrange l small enough, eventually subdividing the cone, so that the following
two estimates hold with uniform constants C1 , C2 > 0, which do not depend on x1 .
Let t̄ − σ < τ < t̄ − σ + l, let A be a Borel set in (Cn+1 (t̄, x̄))t , for t in [τ, t̄ − σ + l]. Then,
for any 0 ≤ δ ≤ t − τ and every set Ax1 , for x1 ∈ I, we have
x1
Hn−1 (X̄t,τ
(Ax1 )) ≥ C1 Hn−1 (Ax1 ) − (t − τ )C2 µ̄x1 (t, Ax1 ),
67
SBV-like regularity for Hamilton-Jacobi equations: the convex case
(
H
n−1
x1
(X̄t,τ
+δ (Ax1 ))
≥
t − (τ + δ)
t−τ
)n−1
x1
Hn−1 (X̄t,τ
(Ax1 )).
x1
Here the (n − 1)-dimensional map X̄t,τ
(x̂) is defined
x1
X̄t,τ
(x̂) := x̂ − (t − τ )d¯x1 (t, x̂).
Consider now the vector field d.
On the set Cn+1 (t̄, x̄) ⊂ U , the matrix of Radon measures Dx d has no jump part. Moreover,
since (Cn+1 (t̄, x̄))t is contained on the set {x| d1 (t, x) = 0} and d(t, ·) is BV, Proposition 1.10
implies
∂
d1 (t, (Cn+1 (t̄, x̄))t ) = 0 for j = 1, . . . , n.
∂xj
Therefore
ˆ ·)
divd(t, ·) = divn−1 d(t,
on (Cn+1 (t̄, x̄))t ,
ˆ x) := (d2 (t, x), . . . , dn (t, x)).
d(t,
For every (t, x) ∈ Cn+1 (t̄, x̄), ux1 (t, x2 ) = u(t, (x1 , x2 )) implies
ˆ x) = dx (t, x̂).
d(t,
1
ˆ x1 , ·), being a (n − 1)-dimensional section of the BV vector field d(t, ·),
The vector field d(t,
belongs, for Proposition 1.11, to [BV ((Σn−1 )x1 )]n−1 for H1 -a.e. x1 such that (Σn−1 )x1 is non
empty.
Since even d¯x1 (t, ·) is BV on (Cn+1 (t̄, x̄))t , Proposition 1.9 implies
ˆ (x1 , ·)) = divn−1 d¯x (t, ·)
divn−1 d(t,
1
for almost every x1 such that (Σn−1 )x1 is non empty. Therefore taken a Borel set A ⊂
(Cn+1 (t̄, x̄))t and any ϕ ∈ Cc∞ ((Cn+1 (t̄, x̄))t ),
∫
∫ ∫
ϕ(x)dµ(t, x) =
ϕ(x)dµ̄x1 (t, x̂)dx1 .
A
I
Ax1
Moreover, for every x ∈ (Cn+1 (t̄, x̄))t
x1
Xt,τ (x) = x − (t − τ )d(t, x) = (x1 , X̄t,τ
(x̂)).
The uniformity on every Ax1 allow us to integrate with respect to H1 , over the set I, to obtain
the following estimates.
Let t̄ − σ < τ < t, let A be a Borel set in (Cn+1 (t̄, x̄))t , for t in [t̄ − σ, t̄ − σ + l]. Then for
any 0 ≤ δ ≤ t − τ , it holds
Hn (Xt,τ (A)) ≥ C1 Hn (A) − (t − τ )C2 µ(t, A),
(
H (Xt,τ +δ (A)) ≥
n
t − (τ + δ)
t−τ
68
)n−1
Hn (Xt,τ (A)).
4.4 The multi-dimensional case
Therefore, repeating the standard procedure seen in Subsection 4.2, we can prove that
µ(t, ·) := divd(t, ·) has Cantor part only for a countable number of t’s in [t̄ − σ, t̄ − σ + l].
Finally, using again Besicovitch Theorem, the set Σcn−1 can be fully covered by a countable
i
number of cones Cn+1
for i ∈ N with the property stated above. Thus the Radon measure
divd(t, ·) can have Cantor part on (Σcn−1 )t only for a countable number of t’s in [0, T ].
We iterate the procedure subdividing Σj−1 in Σj−2 and Σcj−2 . Hence to prove Claim 2.(j-1)
is enough to prove Claim 2.(2), i.e. for j = 3.
Claim 2.(2) The Radon measure divd(t, ·), restricted to (Σ2 )t , can have Cantor part only
for a countable number of t’s in [0, T ].
Proof. The proof is equal to the one done in the 2-dimensional case. We rewrite it with the
notation which applies in this case.
First, suppose V2 is a single straight line. Without loss of generality we can fix
V2 = {v ∈ Rn | v1 = 0, . . . , vn−1 = 0}.
Recall that V2 is a straight line in V3 = {v ∈ Rn | v1 = 0, . . . , vn−2 = 0}.
Call LV2 : R → R the restriction of the Lagrangian LV3 to V2 ,
LV2 (vn ) := LV3 (0, vn ) = L(0, . . . , 0, vn )
for any vn ∈ R. For i = 1 . . . , n − 1, call Ii ⊂ R the set of every xi in R such that (Σ2 )xi is non
empty and I := I1 × · · · × In−1 ⊂ Rn−1 .
For every (x1 , . . . , xn−1 ) ∈ I, we consider the one-dimensional Hamilton-Jacobi equation for
the function ux1 ,...,xn−1 (t, xn ).
{
∂t ux1 ,...,xn−1 + HV2 (Dxn ux1 ,...,xn−1 ) = 0 in (Σ2 )x1 ,...,xn−1 ,
ux1 ,...,xn−1 (0, xn ) = u(0, (x1 , . . . , xn )) ∀xn ∈ ((Σ2 )x1 ,...,xn−1 )0 ,
where HV2 (pn ) is the Hamiltonian associated to LV2 (vn ).
The viscosity solution ux1 ,...,xn−1 (t, xn ) is equal to u(t, (x1 , . . . , xn )) for (t, (x1 , . . . , xn )) ∈ Σ2 .
Indeed
{
(
)}
xn − yn
ux1 ,...,xn−1 (t, xn ) = min u(0, (x1 , . . . , xn−1 , yn )) + tLV2
= u(t, (x1 , . . . , xn )),
yn ∈R
t
where the last equality follows from the fact that, for (t, x) in Σ2 , the unique minimizer in (1.10)
is y = (x1 − td1 (t, x), . . . , xn − tdn (t, x)) and d(t, x) = (0, . . . , 0, dn (t, x)) on Σ2 .
Let us define as usual
dx1 ,...,xn−1 (t, xn ) := (HV2 )pn (Dxn ux1 ,...,xn−1 (t, xn )),
and
µx1 ,...,xn−1 (t, ·) :=
∂
dx ,...,xn−1 (t, ·).
∂xn 1
The vector field dx1 ,...,xn−1 (t, ·) is one-dimensional. Hence, for Theorem 4.2, dx1 ,...,xn−1 (t, ·)
belongs to BV (((Σ2 )x1 ,...,xn−1 )t ) for any (x1 , . . . , xn−1 ) ∈ I, for any t ∈ [0, T ].
69
SBV-like regularity for Hamilton-Jacobi equations: the convex case
On the set Σ2 ⊂ U , the matrix of Radon measures Dx d has no jump part. Moreover, since
(Σ2 )t is contained on the set {x| d1 (t, x) = 0, . . . , dn−1 (t, x) = 0} and d(t, ·) is BV, Proposition
1.10 implies
∂
di (t, (Σ2 )t ) = 0 for i = 1, . . . , n − 1 and l = 1, . . . , n.
∂xl
Therefore
divd(t, ·) =
∂
dn (t, ·)
∂xn
on (Σ2 )t .
For every (t, x) ∈ Σ2 , ux1 ,...,xn−1 (t, xn ) = u(t, (x1 , . . . , xn )) implies
dn (t, x) = dx1 ,...,xn−1 (t, xn ).
The vector field dn (t, (x1 , . . . , xn−1 , ·)) is a one-dimensional restriction of dn (t, ·) thus, for Proposition 1.11, belongs to BV (((Σ2 )x1 ,...,xn−1 )t ) for almost every (x1 , . . . , xn−1 ) ∈ I. Since even
dx1 ,...,xn−1 (t, ·) is BV on ((Σ2 )x1 ,...,xn−1 )t , Proposition 1.9 implies
∂
∂
dn (t, (x1 , . . . , xn−1 , ·)) =
dx ,...,xn−1 (t, ·)
∂xn
∂xn 1
for Hn−1 -a.e. (x1 , . . . , xn−1 ) ∈ I . Therefore taken a Borel set A ⊂ (Σ2 )t and any ϕ ∈ Cc∞ ((Σ2 )t ),
∫
∫ ∫
ϕ(x)dµ(t, x) =
ϕ(x)dµx1 ,...,xn−1 (t, xn )d(x1 , . . . , xn−1 ).
A
I
Ax1 ,...,xn−1
Thanks to the convexity of LV2 , we can apply Theorem 4.4 to µx1 ,...,xn−1 (t, ·) and obtain the
following estimates.
For any τ > 0, let A be a Borel set in (Σ2 )t , t ∈ (τ, T ]. Then for any 0 ≤ δ ≤ t − τ and every
section Ax1 ,...,xn−1 , for (x1 , . . . , xn−1 ) ∈ I, we have
x ,...,xn−1
H1 (Xt,τ1
(Ax1 ,...,xn−1 )) ≥ H1 (Ax1 ,...,xn−1 ) − (t − τ )µx1 ,...,xn−1 (t, Ax1 ,...,xn−1 ),
x ,...,xn−1
H1 (Xt,τ1 +δ
(Ax1 ,...,xn−1 )) ≥
x ,...,xn−1
Here we denote with Xt,τ1
(xn ) the one-dimensional map defined on ((Σ2 )x1 ,...,xn−1 )t
x ,...,xn−1
Xt,τ1
t − (τ + δ) 1 x1 ,...,xn−1
H (Xt,τ
(Ax1 ,...,xn−1 )).
t−τ
(xn ) := xn − (t − τ )dx1 ,...,xn−1 (t, xn ).
The corresponding n-dimensional map defined on (Σ2 )t
Xt,τ (x) := x − (t − τ )d(t, x),
reduces to
x ,...,xn−1
Xt,τ (x) = (x1 , . . . , xn−1 , Xt,τ1
for every x ∈ (Σ2 )t .
70
(xn ))
4.4 The multi-dimensional case
We can integrate the previous estimates with respect to Hn−1 over I to recover estimates of
type (4.1) and (4.2). For any τ > 0, given a Borel set A ⊂ (Σ2 )t , for t in [τ, T ], we have
Hn (Xt,τ (A)) ≥ Hn (A) − (t − τ )µ(t, A).
(4.9)
For any τ > 0, given a Borel set A ⊂ Σt , for t in [τ, T ] and 0 ≤ δ ≤ t − τ we have
Hn (Xt,τ +δ (A)) ≥
t − (τ + δ) n
H (Xt,τ (A)).
t−τ
(4.10)
Thus the strategy seen in the Subsection 4.2 can be easily applied to prove that µ(t, ·),
restricted to (Σ2 )t , can have Cantor part only for a countable number of t’s in [0, T ].
Consider now the case in which V2 consists of a finite number of straight lines. When we
consider µ(·, ·) restricted to the points of Σ2 such that d(t, x) belongs only to a part of one of
the straight lines, we can apply the considerations done in the case where V2 consists only of a
single straight line. On the other hand, when we consider µ(·, ·) restricted to the points of Σ2
such that d(t, x) belongs to an intersection point of two, or more, straight lines, the divergence
divd(t, ·) = µ(t, ·) must be zero on every Borel set, as seen in the 2-dimensional case. The case
in which V2 is contained in a finite number of straight lines is analogous.
Thus the measure µ(t, ·) can have Cantor part only for a countable number of t’s in [0, T ]
even when V2 consists of a finite number of straight lines.
Once Claim 2.(2) is proved, we can iteratively prove all the others Claims 2.(j-1) for j =
4, · · · , n just by repeating the same considerations for the general case in which Vj consists of a
finite union of (j − 1)-dimensional planes. This case can be treated as usual distinguishing the
two cases. When we consider µ(·, ·) restricted to the points of Σj such that d(t, x) belongs only
to a part of one of the (j − 1)-dimensional planes, we can apply the considerations done in the
case where Vj consists only of a single (j − 1)-dimensional plane. On the other hand, when we
consider µ(·, ·) restricted to the points of Σj such that d(t, x) belongs to a (j − 2)-dimensional
plane intersection of two, or more, (j − 1)-dimensional planes, we can reduce the problem to the
(j − 2)-dimensional case. Indeed in this case we can apply again the iterative proof. The case
in which Vj is contained in a finite number of (j − 1)-dimensional planes is analogous.
The considerations above done for j = n + 1 concludes even the proof of Claim 2.(n).
Let us recall all the necessary assumptions.
Suppose H is C 2 (Rn ) convex and
H(p)
= +∞.
|p|→∞ |p|
lim
(HYP(0)) The vector field d(t, ·) belongs to [BV (Ωt )]n for every t ∈ [0, T ].
Define Vπn as
Vπn := {v ∈ Rn | L(·) is not twice differentiable in v},
and
Σπn := {(t, x) ∈ U | d(t, x) ∈ Vπn }
71
and Σcπn := U \ Σπn .
SBV-like regularity for Hamilton-Jacobi equations: the convex case
(HYP(n)) We suppose Vπn to be contained in a finite union of hyperplanes Ππn .
For j = n, . . . , 3 for any (j − 1)-dimensional plane πj−1 in Ππj , let Lπj−1 : Rj−1 → R be the
(j − 1)-dimensional restriction of L to πj−1 and
Vπj−1 := {v ∈ Rj−1 | Lπj−1 (·) is not twice differentiable in v}.
Define
Σπj−1 := {(t, x) ∈ Σπj | d(t, x) ∈ Vπj } and
Σcπj−1 := Σπj \ Σπj−1 .
(HYP(j-1)) We suppose Vπj−1 is contained in a finite union of (j − 2)-dimensional planes Ππj−1 ,
for every πj−1 ∈ Ππj .
Remark 4.7. There is no need to ask any assumption on the one-dimensional restriction of L
to a straight line in any of the Vπ2 for a plane π2 , since in the one-dimensional case the SBV
regularity is proven without any further assumptions on L.
Theorem 4.8. With the above assumptions (HYP(0)),(HYP(n)),...,(HYP(2)), the Radon measure divd(t, ·) has Cantor part on Ωt only for a countable number of t’s in [0, T ].
The following corollaries are easily obtained from Theorem 4.8.
Corollary 4.9. Let Dx u(t, ·) belongs to [BV (Ωt )]n for every t ∈ [0, T ] and let L satisfy the
assumptions (HYP(n)),. . . , (HYP(2)), then the Radon measure divd(t, ·) has Cantor part on Ωt
only for a countable number of t’s in [0, T ].
Proof. If Dx u(t, ·) belongs to [BV (Ωt )]n for every t ∈ [0, T ], then d(t, ·) = Hp (Dx u(t, ·)) belongs
to [BV (Ωt )]n for every t ∈ [0, T ].
Corollary 4.10. Let u(0, ·) be semiconcave and let L satisfy (HYP(n)),. . . , (HYP(2)), then the
Radon measure divd(t, ·) has Cantor part on Ωt only for a countable number of t’s in [0, T ].
Proof. It follows from Proposition 1.35.
72
Chapter 5
Some applications
In this chapter we present some simple applications of Ambrosio and De Lellis’s SBV regularity
theorem 0.1, for entropy solutions of the one-dimensional scalar conservation laws
in Ω := R+ × (a, b).
∂t U + Dx (H(U )) = 0
(5.1)
That theorem can be easily extended to one-dimensional Hamilton-Jacobi equations. Indeed,
the potential, given by
{
∂t u = −H(U )
Dx u = U,
is a viscosity solution of the Hamilton-Jacobi equation
∂t u + H(Dx u) = 0
in Ω
(5.2)
if and only if U is an entropy solution to (5.1). Therefore, Theorem 0.1 applies also to the
distributional derivative of a viscosity solution of Hamilton-Jacobi equation (5.2) when H is
C 2 (Ω) and locally uniformly convex.
In Sections 5.1 and 5.2 we describe Generalized Hydrostatic Boussinesq (GHB) equations
and the model of sticky particles, then, in Section 5.3, we show how Theorem 0.1 of Ambrosio
and De Lellis (whose proof can be found in [4]) applies to them in the one-dimensional case. In
the last section we present a counterexample which prevent us from using the same approach
for the multi-dimensional case. A similar counterexample was shown by Vasseur in [38], but it
was never published.
The results presented here can be found in Tonon [37].
5.1
Generalized Hydrostatic Boussinesq equations
Generalized Hydrostatic Boussinesq (GHB) equations can be seen as the most degenerate version
of Generalized Navier-Stokes Boussinesq (GNSB) equations, where both the inertia terms and
the dissipative operator are neglected. These equations rule the dynamic of a fluid under fast
convection. In terms of the temperature of the fluid they take the form
y = x + ∇p,
∇ · v = 0,
73
Some applications
∂t y + (v · ∇)y = G(x),
(5.3)
Rn
here, being D ⊂
a smooth bounded domain where the fluid is placed, the function y(t, x) :
R+ × D → Rn is the generalized temperature field of the fluid, v(t, x) : R+ × D → Rn its
velocity, p(t, x) : R+ × D → R the pressure, G(x) : Rn → Rn the generalized heat source term,
an [L∞ (Rn )]n function. Equation (5.3) can be seen as a generalization of the hydrostatic balance
in Convection Theory.
The fact that G depends only on the position of the fluid allows us to apply Theorem 0.1.
However, this is a very particular assumption since the heat source can depend also on time and
temperature G = G(t, x, y).
The complete description of this system can be found in [17], Chapter 3. Passing to Lagrangian coordinates, Brenier proved there that a generalized solution can be constructed.
Since we need some concepts of Optimal Transport Theory let us recall some preliminary
definitions and results.
First, we introduce rearrangements and measure preserving maps.
Given two [L2 (D)]n
maps Y and Z, we say that they are rearrangement of each other if they define the same image
measure, i.e. for all continuous f on Rn , such that |f (y)| ≤ 1 + |y|2 ,
∫
∫
f (Y (a))da =
f (Z(a))da.
D
D
[L2 (D)]n
We say that Y in
is a measure preserving map, when it is a rearrangement of the
identity map, i.e. for all continuous f on Rn , such that |f (y)| ≤ 1 + |y|2 ,
∫
∫
f (a)da =
f (Y (a))da.
D
D
Next we define the class of maps with convex potential. We say that an [L2 (D)]n map Y
belongs to the class C of maps with a convex potential, if there is a lower semi-continuous convex
function p : Rn → (−∞, +∞] such that, for Hn -a.e. point x in D, the gradient ∇p(x) coincides
with Y .
Then, looking for a rearrangement with convex potential, we have the following Brenier’s
Theorem which can be found in [14]:
Theorem 5.1 (Brenier). Let Y be a non degenerate [L2 (D)]n map. Then there is a unique
polar factorization
Y = Y R ◦ X,
where Y R belongs to C and X is a Lebesgue measure preserving map of D.
In this decomposition, Y R is the unique rearrangement of Y in C and X is the unique
measure preserving map of D that minimizes
∫
|X(a) − Y (a)|2 da.
D
In addition, X can be written:
X(a) = (∇ϕ)(Y (a)),
Hn -a.e. a ∈ D,
where ϕ is a convex Lipschitz function defined on Rn .
74
5.1 Generalized Hydrostatic Boussinesq equations
Coming back to our problem and passing to Lagrangian coordinates the system (5.1,5.3)
looks like
Y (t, a) = X(t, a) + ∇p(t, X(t, a)),
(5.4)
∂t Y (t, a) = G(X(t, a)),
(5.5)
where, for all t, D ∋ a 7→ X(t, a) is a measure preserving map as a consequence of the fact that
v is a smooth divergence-free vector field. X(t, a) denotes the position of a fluid particle a at a
time t, therefore its velocity and its temperature are
∂t X(t, a) = v(t, X(t, a)),
Y (t, a) = y(t, X(t, a)).
Note that, due to the above equations, if particles reach the same position they will have
the same velocity, the same temperature and will no more separate. In fact GHB equations are
very closed to systems of sticky particles as we can see in the next section.
Assuming a priori that the map x 7→ x + ∇p(t, x) has convex potential, we deduce from (5.4)
that x 7→ x + ∇p(t, x) is the unique convex rearrangement Y R (t, ·) of Y (t, ·), due to the fact
that Y (t, a) = Y R (t, ·) ◦ X(t, a). Moreover (5.5) implies that for all C 1 function f compactly
supported on Rn , Y R (t, ·) satisfies
∫
∫
d
R
(5.6)
f (Y (t, a))da =
(∇f )(Y R (t, a)) · G(a)da.
dt D
D
With these considerations Brenier naturally introduced a more general concept of solution
to GHB system. We say that Y CR in C 0 ([0, T ], [L2 (D)]n ) is the convex rearrangement (CR)
solution to the GHB equations (5.4, 5.5), if :
• Y CR (t, ·) belongs to the set C of all maps with convex potential, for all t ∈ [0, T ],
• for all compactly supported C 1 function f on Rn , Y CR (t, ·) satisfies (5.6).
In [17], he proved the following existence theorem.
Theorem 5.2 (Brenier). For each initial condition Y 0 in [L2 (D)]n , there is at least one CRR
solution Y CR (t, a) such that Y CR (0, ·) = (Y 0 ) (·).
This solution can be obtained as the limit in C 0 ([0, T ], [L2 (D)]n ) as h → 0, of a time discrete
approximation Y h (t, a) defined, first at discrete times t = nh, by:
R
Y h (nh + h, a) = [Y h (nh, a) + hG(a)] ,
n = 0, 1, 2, . . .
(where, as seen before, (·)R is the convex rearrangement operator) and then linearly interpolated
in t.
The time discrete approximation, given by the theorem above, tells us that starting from an
R
initial temperature datum Y 0 , the CR-solution evolves linearly as (Y 0 ) (a) + tG(a) as far as
this function remains with convex potential. When this is no more the case, it is rearranged in
order to preserve the membership to the space of maps with convex potential.
75
Some applications
Note that CR-solutions are Hn -a.e. equal to functions with convex potential, i.e. for all t
there exists a convex function ψt : D → R such that
Y CR (t, a) = Dψt (a),
for Hn -a.e. a in D. Taking now the Legendre transform of this convex function
u(t, x) = sup {x · a − ψt (a)} ,
a∈D
we obtain a function u(t, ·) which is again convex and its distributional derivative Dx u(t, ·) is
the generalized inverse of Y CR (t, ·). We are interested in the regularity of Dx u(t, ·). What we
can say so far is that it is a function of bounded variation.
5.1.1
One-dimensional case
In the one-dimensional case it is possible to look at Dx u(t, ·) as a solution of a scalar conservation
law.
First we can observe that, taking D = [0, 1], the convex rearrangement is the monotone non
decreasing rearrangement defined by
Y R (s) = inf{t ∈ R | µY (t) > s},
for s in [0,1], where
µY (t) = H1 ({Y < t}),
is the distribution function. For a detailed description of monotone non decreasing rearrangement we refer to [34], Chapter 1.
One of the properties of monotone non decreasing rearrangement is that it is non expansive
in L2 ([0, 1]), i.e.
∫
∫
|Y R (a) − Z R (a)|2 da ≤
D
|Y (a) − Z(a)|2 da.
D
This property guarantees the uniqueness of the solution of (5.6).
Moreover, as explained in [16], the limit of the time discrete approximation, defined in
Theorem 5.2, satisfies the sub-differential inclusion:
G(x) ∈ ∂t Y + ∂Ψ[Y ],
where Ψ[Y ] = 0 if Y is a non decreasing function of x in D, and Ψ[Y ] = +∞ otherwise.
The generalized inverse of the solution, in the one-dimensional case, can be found using the
Heaviside function. Looking at its behavior, Brenier proved in [15], the following theorem. In the
proof he used a Transport Collapse method, which involves the same time discrete approximation
scheme seen in Theorem 5.2.
Theorem 5.3 (Brenier). Let Y CR (t, a) be the CR-solution found in Theorem 5.2, then the
generalized inverse
∫
1
U (t, y) =
H(y − Y CR (t, a))da,
0
76
5.2 Sticky particles
where H is the Heaviside function, is an entropy solution of the scalar conservation law
∂t U + Dx (H(U )) = 0,
where H is the primitive of G, Hp (p) = G(p).
We are interested in the regularity of an entropy solution of a scalar conservation law with
non decreasing initial conditions and Lipschitz flux function H. Applying what we have already
said, since we are in the one-dimensional case, the entropy solution above can be seen as the
derivative of the unique viscosity solution of the following Hamilton-Jacobi equation
∂t u + H(Dx u) = 0,
with a convex initial datum and Lipschitz Hamiltonian.
5.2
Sticky particles
At a discrete level pressureless gases with sticky particles can be modeled by a finite collection of particles that get stuck together right after they collide with conservation of mass and
momentum. On the other hand at a continuous level the model is governed by the following
one-dimensional system of conservation laws in (0, +∞) × R
∂t ρ + Dx (ρv) = 0,
∂t (ρv) + Dx (ρv 2 ) = 0,
where ρ(t, x) is the density field, while v(t, x) is the velocity one. This set of equations can be
seen as the limit, when pressure goes to zero, of the usual Euler equations. In [18], Brenier
and Grenier showed that the continuous model can be fully described, in an alternative way, by
scalar conservation laws, with non decreasing initial conditions, general flux functions and the
usual Kruzhkov entropy condition.
In particular they proved that if (ρ, v) is a solution corresponding to sticky particles, then
there exist H ∈ Lip(R) and U entropy solution of
∂t U + Dx (H(U )) = 0,
where U (t, x) = Dx u(t, x) is such that ρ(t, x) = Dx2 u(t, x) is a cumulative distribution function
associated to the probability measure ρ and Hp (p) = v(0, p).
The proof uses a scheme in which a finite number of particles are described by weight, position
and velocity, under the assumption that the speed of a particle is constant as long as it meets no
new particles and it changes only when shocks occur. Only a finite number of shocks can occur
because particles remain stuck together after a collision. Moreover particles having the same
position at a time t move together at the same speed and their total momentum is the sum of their
initial momentum. This scheme is strongly reminiscent of Dafermos’s polygonal approximation
methods for scalar conservation laws, where each particle corresponds to a jump of an entropy
solution of a scalar conservation law with a piecewise linear continuous flux function. Thus, it
77
Some applications
is reasonable to expect, as it is, that the continuous limit of the sticky particles dynamics is
properly described by a scalar conservation law.
The fact that the distribution function U is a non decreasing entropy solution of that scalar
conservation law strictly relates sticky particle system to Convection Theory. Indeed if we take
the generalized inverse of U , which is precisely the monotone rearrangement of the measure ρ,
it turns out that it is exactly the limit of the time discrete approximation seen in Theorem 5.2.
As we did for GHB equations we can relate the non decreasing entropy solutions to the
viscosity solution of an Hamilton-Jacobi equation with convex initial datum.
5.3
Convex solutions of Hamilton-Jacobi equations in the multidimensional case
Let us now consider the following Hamilton-Jacobi equation
∂t u + H(Dx u) = 0,
with initial datum u(0, x) = 12 |x|2 , and Lipschitz Hamiltonian H. Thus we are in a particular
case of the ones considered above. As proved in [6], by Bardi and Evans, the unique viscosity
solution to such an equation has the form
}
{
1 2
|z| + y · (x − z) − tH(y) .
u(t, x) = sup inf
z
2
y
This representation formula is true even in the multi-dimensional case and an analogous one
works as well for general initial datum but convex Hamiltonian. Moreover it is equivalent to
{
}
1 2
u(t, x) = sup x · y − |y| − tH(y) .
(5.7)
2
y
Here the sup becomes a maximum under suitable hypotheses on H.
Note that equation (5.7) is equivalent to saying that u is the Legendre transform of 12 |y|2 +
tH(y). On the other hand since u is, in the GHB equation case, the Legendre transform of
ψt (a), we have the following geometric representation for the CR-solution, for H1 -a.e. a,
Y CR (t, a) = ∇convex(ψ0 (a) + tH(a)),
where convex(f ) = max{g ≤ f | g convex}.
Define
1
v(t, x) := −
t
then
(
)
1 2
u(t, x) − |x| ,
2
{
|x − y|2
v(t, x) = min H(y) +
y
2t
is the unique viscosity solution of
∂t v +
|Dx v|2
=0
2
78
}
5.4 Multi-dimensional case
with Lipschitz initial datum v0 (x) = H(x).
Since the Hamiltonian |x|2 is uniformly convex we can use directly Theorem 0.1 of Ambrosio
and De Lellis, to prove that Dx v(t, ·) belongs to SBV for a.e. t and the same is true also for
Dx u(t, ·).
2
Remark 5.4. From what we have seen, in the one-dimensional case, SBV regularity holds for
the generalized inverse of a solution of the GHB equation with the identity as initial datum and
for the cumulative distribution function associated to the density of the pressureless gas.
5.4
Multi-dimensional case
We wonder if Hamilton-Jacobi equations are a good model for GHB systems or sticky particles
models even in the multi-dimensional case. Are they able to describe the behavior of our
solution? If this was the case we could automatically state SBV regularity applying Theorem
0.2. Unfortunately the answer to our question is negative. In the following subsection we show
a counterexample in which a multi-dimensional solution of an Hamilton-Jacobi equation has a
behavior which is not allowed for GHB systems or sticky particles models, i.e. Theorem 0.2 does
not suit our problem. However, this does not mean that SBV regularity cannot be proved in
some other way.
5.4.1
A counterexample
A first counterexample was found by Vasseur in [38] but it was never published. With that counterexample Vasseur showed a discrepancy between the density distribution ρ̃(t, x) = det(Dx2 u(t, x)),
associated to the solution u of the Hamilton-Jacobi equation ∂t u + H(Dx u) = 0 with initial datum u(0, x) = 12 , and the density distribution ρ(t, x), generated from the identity ρ0 (x) = 1 in
a sticky particles process with speed v = Hp (p). Indeed, he proved the existence of a time t at
which the two density distribution differ.
The following counterexample shows the same discrepancy, underlining in addiction the cause
of it. Hamilton-Jacobi equations allow separations of particles after collisions.
Consider the viscosity solution of the Hamilton-Jacobi equation
1
vt + |Dx v|2 = 0,
2
in R2 , with initial datum

|x|2

 − 2 for x ∈ B(0, 1)
v(0, x) = f (x) for x ∈ B(0, 2) \ B(0, 1)

 −|x | for x ∈ R2 \ B(0, 2),
1
where f (x) joins smoothly − |x|2 to −|x1 | and satisfies f (x) > − |x|2 in B(0, 2) \ B(0, 1), being
B(x, r) the open ball with center in x and radius r > 0.
2
2
79
Some applications
Note that following upside down the passages seen in Section 5.3 we can recover from v a
convex viscosity solution of the equation
∂t u + H(Dx u) = 0,
u(0, x) =
|x|2
2
where
|x|2
2
and H(x) = v(0, x) is a smooth function. We are thus considering a viscosity solution of
Hamilton-Jacobi with convex initial datum. If Hamilton-Jacobi were the good model for GHB
and sticky particles systems, passing to the Legendre transform of our viscosity solution we
should recover the CR-solution limit of the time-discrete approximation scheme.
Using the Hopf-Lax formula for convex Hamiltonians we recover the viscosity solution for
any time t
{
}
(x1 − y1 )2 + (x2 − y2 )2
v(t, x) = min v(0, y) +
.
y
2t
u(t, x) = −tv(t, x) +
Let us compute the value of v for t = 1 in the origin:
{
}
y12 + y22
v(1, (0, 0)) = min v(0, y) +
.
y
2
(5.8)
Observe that for any y in B(0, 1) we have
v(0, y) +
y12 + y22
= 0.
2
v(0, y) +
y12 + y22
> 0.
2
For any y in B(0, 2) \ B(0, 1)
For any y in R2 \ B(0, 2) we have
v(0, y) +
y12 + y22
y 2 + y22
y2
= −|y1 | + 1
≥ −|y1 | + 1 ,
2
2
2
hence we can restrict the minimum in the region R2 \ B(0, 2) to points with y2 = 0, moreover
for that points we have
y2
−|y1 | + 1 ≥ 0,
2
and the equality occurs only for (−2, 0) and (2, 0).
Thus the minimum in (5.8) is obtained if and only if y belongs to the set B(0, 1)∪{(−2, 0), (2, 0)}.
The origin is therefore a point of non differentiability for v(1, ·) with the convex hull of the set of
all minima as superdifferential. This means that all the points in the set B(0, 1)∪{(−2, 0), (2, 0)},
which is of positive H2 -measure, are transported by the flux along straight line trajectories which
collide at time t = 1 in the position (0, 0).
80
5.4 Multi-dimensional case
However, for any δ > 0, we have to compute
{
}
y12 + y22
v(1 + δ, (0, 0)) = min v(0, y) +
.
y
2(1 + δ)
(5.9)
For any y in B(0, 1) we have
v(0, y) +
y12 + y22
y 2 + y22
δ
= −δ 1
≥−
.
2(1 + δ)
2(1 + δ)
2(1 + δ)
For any y in B(0, 2) \ B(0, 1)
v(0, y) +
y12 + y22
y 2 + y22
4δ
> −δ 1
>−
.
2(1 + δ)
2(1 + δ)
2(1 + δ)
4δ
δ
Here we note that − 2(1+δ)
< − 2(1+δ)
.
2
For any y in R \ B(0, 2) we have
v(0, y) +
y12 + y22
y 2 + y22
y12
= −|y1 | + 1
≥ −|y1 | +
,
2(1 + δ)
2(1 + δ)
2(1 + δ)
hence we can restrict the minimum to the points in R2 \ B(0, 2) with y2 = 0. Moreover, for that
points we have that the minimum value is reached for |y1 | = 2 if 1 + δ < 2, for |y1 | = 1 + δ
otherwise.
In the first case
4δ
y12
=−
.
−|y1 | +
2(1 + δ)
2(1 + δ)
In the second one
−|y1 | +
(1 + δ)2
4δ
y12
=−
<−
.
2(1 + δ)
2(1 + δ)
2(1 + δ)
Thus (0,0) is a point of non differentiability even for t = 1 + δ for any δ > 0. Moreover its
superdifferential is the set [(−2, 0), (2, 0)] for 0 < δ < 1, or the set [(−(1 + δ), 0), ((1 + δ), 0)]
for δ > 1. In any case it is a set of positive H1 -measure. This set has non empty intersection
with the superdifferential of v in the origin at time t = 1 but does not contain the whole
of it. Recall that, the superdifferential of v in the origin contains, at time t = 1, the set
convex(B(0, 1) ∪ {(−2, 0), (2, 0)}) which is a set of positive H2 -measure.
Points, being in B(0, 1) ∪ {(−2, 0), (2, 0)} at time t = 0, collide at time t = 1 and separate
at time t = 1 + δ for any δ > 0.
We have thus shown an example of a viscosity solution in which a point of non differentiability
of zero codimension evolves in a point of non differentiability of codimension one.
2
As we have already said, coming back to u(t, x) = −tv(t, x)+ |x|2 and passing to the Legendre
transform of our viscosity solution, we should obtain the CR-solution of the GSB equation.
However for this function a flat part of dimension two would evolve in a flat part of dimension
one, in contrast with propagation of flat parts. Particles stuck together could have different
velocities but this is not the case for GHB and the sticky particles model.
Hence GHB and the sticky particles model cannot be truly described by Hamilton-Jacobi
equation in the multi-dimensional case.
81
Chapter 6
Decomposition of BV functions
The aim of this chapter is to give a generalization of Jordan decomposition property to real
valued BV functions of many variables.
The starting point is a recent result presented to us by Alberti, Bianchini and Crippa, which
shows that a real Lipschitz function of many variables with compact support can be decomposed
in sum of monotone functions. Precisely they give the following definition of monotone function
Definition 6.1. A function f : Rn → R, which belongs to Lip(Rn ), is said to be monotone if
the level sets {f = t} := {x ∈ Rn | f (x) = t} are connected for every t ∈ R.
and state the following theorem.
Theorem 6.2 (Alberti, Bianchini and Crippa). Let f be a function in Lipc (Rn ) with compact
support. Then there exists a countable family {fi }i∈N of functions in Lipc (Rn ) such that f =
∑
i fi and each fi is monotone. Moreover there is a pairwise disjoint partition {Ωi }i∈N of Borel
sets of Rn such that ∇fi is concentrated on Ωi .
In the case of BV functions, which are defined Hn -a.e., an appropriate generalization of the
concept of monotone function has to involve super-level sets, sub-level sets and the concept of
indecomposable set, as given in [3].
Definition 6.3. A set E ⊆ Rn with finite perimeter is said to be decomposable if there exists a
partition (A, B) of E such that P (E) = P (A) + P (B) and both Hn (A) and Hn (B) are strictly
positive. A set E is said to be indecomposable if it is not decomposable.
Definition 6.4. A function f : Rn → R, which belongs to L1loc (Rn ), is said to be monotone
if the super-level sets {f > t} := {x ∈ Rn | f (x) > t} and the sub-level sets {f < t} := {x ∈
Rn | f (x) < t} are of finite perimeter and indecomposable for H1 -a.e. t ∈ R.
As proved in Section 6.3, in the case of Lipschitz functions, Definition 6.1 and Definition 6.4
are equivalent.
When comparing the case of functions of one variables with the case of functions of many
variables differences and analogies arise.
83
Decomposition of BV functions
On the one hand, it can be found an L1 monotone function, which is not of bounded variation,
that is a counterexample to the fact that monotonicity is a sufficient condition for being of
bounded variation (Example 6.14).
On the other hand, it can be stated that a BV function is decomposable in a countable sum
of monotone functions, similarly to the case of BV functions of one real variable.
The main result of this chapter is the following.
Theorem 6.5. Let f : Rn → R be a BV (Rn ) function. Then there exists a finite or countable
family of monotone BV (Rn ) functions {fi }i∈I , such that
∑
∑
fi and |Df | =
|Dfi |.
f=
i∈I
i∈I
This decomposition is in general not unique, see Remark 6.12, and it can generate monotone
BV functions without mutually singular distributional derivatives, see Example 6.13. Thus we
loose the property true in the Lipschitz case.
The main tool for proving this theorem is a decomposition theorem for sets of finite perimeter,
presented here in the form given in [3].
Theorem 6.6 (Ambrosio, Caselles, Masnou and Morel). Let E be a set with finite perimeter
in Rn . Then there exists a unique finite or countable family of pairwise disjoint indecomposable
sets {Ei }i∈I such that
∑
Hn (Ei ) > 0 and P (E) =
P (Ei ).
i∈I
Moreover, denoting with
E̊ M :=
{
}
|E ∩ B(x, r)|
x ∈ Rn | lim
=1
|B(x, r)|
r→0+
the essential interior of the set E, it holds
(
)
∪
Hn−1 E̊ M \
E̊iM = 0
i∈I
and the Ei ’s are maximal indecomposable sets, i.e. any indecomposable set F ⊆ E is contained,
up to Hn -negligible sets, in some set Ei .
The chapter is organized as follows.
In Section 6.1 we prove the decomposition theorem for Lipschitz functions.
In Section 6.2 we generalize the decomposition theorem to BV functions and show that this
decomposition can generate monotone BV functions without mutually singular distributional
derivatives.
In Section 6.3 we give two counterexamples: the first to the fact that a monotone function
is always a BV function, the second to a further extension of the Theorem 6.5 to vector valued
functions. We also give a proof of the fact that for Lipschitz functions Definition 6.1 and
Definition 6.4 are equivalent.
84
6.1 The Decomposition Theorem for Lipschitz functions from Rn to R
6.1
The Decomposition Theorem for Lipschitz functions from
Rn to R
Before proving the decomposition theorem for Lipschitz functions we state some results on the
structure of their level sets.
We first set some notations.
Let f : Rn → R belong to Lipc (Rn ). For every t ∈ R we call Et := {x| f (x) = t}, we denote
with Ct the family of all connected components C of Et such that Hn−1 (C) > 0 and we denote
with Etc the union of all C in Ct .
Theorem 6.7. Let f : Rn → R belong to Lipc (Rn ) and have compact support. Then for almost
every t ∈ R
i) Et is Hn−1 -rectifiable and Hn−1 (Et ) < +∞;
ii) the map f is differentiable in x for Hn−1 -a.e. x ∈ Et ;
iii) the family Ct of open connected components of Et is countable and Hn−1 (Et \ Etc ) = 0.
Proof. We refer to Theorem 2.5 in [2].
Lemma 6.8. Let f : Rn → R be a Lipschitz function with compact support. Then the set Etc
for any t ∈ R and the set E c := ∪t∈R Etc are a countable union of closed sets in Rn ; in particular
they are Borel measurable.
Proof. We refer to Lemma 6.1 in [2].
We show now the proof of Theorem 6.2 as presented to us by Alberti, Bianchini and Crippa.
Proof of Theorem 6.2. We divide the proof in several steps.
Step 1. Assume that f ≥ 0 and that f has a strictly positive maximum. Take a countable dense
sequence of 0 < ti ∈ R+ , such that the conclusions of Theorem 6.7 hold. For 0 < ti < max f , let
Gi be the connected unbounded open component of {f < ti } := {x ∈ Rn | f (x) < ti }, note that
due to the fact that f has compact support there can be only one of such components. Let Fi
be the compact set Rn \ Gi , and decompose it into the connected compact components Fij with
positive Hn−1 -measure. At least one of such components exists thanks to the choice of the ti ’s.
It is clear that for ti ≤ ti′
Gi ⊆ Gi′ ,
1
dH (Fi , Fi′ ) ≥ |ti − ti′ |,
c
(6.1)
the first holds because Gi ⊂ {f < ti′ } and Gi is unbounded thus Gi ⊂ Gi′ , the second follows
from the Lipschitz estimate, when dH is the Hausdorff distance.
Moreover Gij = Rn \ Fij is open and connected: in fact, is the complement of a closed set
and it can be written as the union of Gi with the neighborhoods of each connected components
C ⊂ Et \ Fij not intersecting Fij .
85
Decomposition of BV functions
Step 2. Define the following partial order relation on the countable family Fij as
Fij ≤ Fi′ j ′
if
ti ≤ ti′ , Fij ⊇ Fi′ j ′ .
Let Fij , i ∈ I, j ∈ J be a maximal countable ordered sequence. Note that we do not need the
Axiom of Choice here because the family is countable.
From the definition of partial order the index j must be a function of i: j = j(i), i ∈ I.
Since the sequence must be maximal the sequence ti is dense in a segment [0, t̄ ], by (6.1). Here
t̄ := max f .
Step 3. Define the function f˜ by
{
}
f˜(x) := sup ti | x ∈ Fij(i) , i ∈ I .
By the definition of f˜, f˜(x) − f˜(y) ≥ k implies that for all ε > 0 there are i, i′ ∈ I such that
y∈
/ Fij(i) , x ∈ Fi′ j(i′ ) and ti′ − ti > k − ε. Hence
|f˜(x) − f˜(y)| ≤ |ti − ti′ | ≤ c|x − y|.
Thus f˜ is c-Lipschitz. Since the sets Fij are uniformly bounded, f˜ has bounded support and
max f˜ = t̄ = max f .
Step 4. We observe that for each 0 ≤ ti ≤ t̄ we have {f˜ ≥ ti } = Fi,j(i) .
Moreover for 0 ≤ ti < ti′ ≤ t̄ the set
{ti ≤ f˜ < ti′ } = Fij(i) \ Fi′ j(i′ ) = Fij(i) ∩ Gi′ j(i′ )
is arc connected, because Gi′ j(i′ ) is open connected, hence arc connected, and dH (Gi′ j(i′ ) , Fij(i) ) >
0.
It follows that its closure {ti ≤ f˜ ≤ ti′ } is compact connected, and the intersection as ti ↗ t,
ti ↘ t is connected.
The case {0 ≤ f < ti } = ∩ti ↘0 Gij(i) can be treated similarly because of the ordering of Gij(i)
and the compactness connectedness of Gij(i) ∩ B(0, R), for R ≫ 1.
Therefore f˜ is monotone.
Step 5. We now use the fact that for each i, j one has ∂Fij ⊂ Etci by construction. Let
Ẽt := {f˜ = h} be a level set with empty interior: hence each x ∈ Ẽt is the limit of a sequence
of points in ∪i ∂Fij(i) , and by the continuity of f , f˜ we conclude that f = f˜ on Ẽt .
In particular if Ẽ c is defined as in Lemma 6.8 for the function f˜, then Ẽ c ∩ Et = Ẽt , and by
the Coarea Formula
∫
∫
∫
∫
n
n−1
c
1
n−1
c
1
∇f (x)dH (x) =
H
(Ẽ ∩ Et )dH (t) =
H
(Ẽ ∩ Ẽt )dH (t) =
∇f˜(x)dHn (x).
Ẽ c
R
R
Ẽ c
We conclude thus that ∇f = ∇f˜ Hn -a.e. on Ẽ c .
Step 6. Since ∇f˜ = 0 Hn -a.e. on Rn \ Ẽ c , we conclude that f ′ = f − f˜ is again c-Lipschitz, but
its total variation is diminished by the total variation of f˜.
86
6.2 The Decomposition Theorem for BV functions from Rn to R
Since the
there is at most a countable family of fi ̸= 0 such
∑ total variation of f is bounded,
c
that f = i fi , and if we denote with E (i) the sets defined in Lemma 6.8 for the Lipschitz
function fi , then E c (i) \ E c (j) is still a Borel set. Moreover, ∇fj = 0 Hn -a.e. on E c (i) for i ̸= j,
so that fj ♯ Hn xE c (i) is singular w.r.t. H1 .
The proof is complete by defining Ωi := E c (i) \ (∪j<i E c (j)).
6.2
The Decomposition Theorem for BV functions from Rn to
R
To generalize the Jordan decomposition property, let us concentrate on functions f : Rn → R,
which belong to BV (Rn ). From now on n > 1.
Since we will consider functions of bounded variation, the Definition 6.4 of monotone function
becomes the following:
Definition 6.9. A BV function f : Rn → R is said to be monotone if the super-level sets
{f > t} = {x ∈ Rn | f (x) > t} and the sub-level sets {f < t} = {x ∈ Rn | f (x) < t} are
indecomposable, for H1 -a.e. t ∈ R.
Indeed, we recall that, for BV functions, super-level sets and sub-level sets are of finite
perimeter for H1 -a.e. t ∈ R.
We now prove the main theorem of this chapter.
Proof of Theorem 6.5. The proof will be given in several steps.
Before entering into details, let us consider the following simple case.
Let f = χE with E ⊆ Rn a decomposable set of finite perimeter such that Rn \ E is indecomposable. Thanks to Theorem 6.6, there exists a unique finite or countable family of pairwise
disjoint indecomposable sets {Ei }i∈I such that
∑
Hn (Ei ) > 0 and P (E) =
P (Ei ).
i∈I
To see the properties of Rn \ Ei let us consider the following lemma.
Lemma 6.10. Let E be a decomposable set of finite perimeter such that Rn \E is indecomposable.
Let {Ei }i∈I be the family of its indecomposable components given by Theorem 6.6. Then Rn \ Ei
is indecomposable for every i ∈ I.
Proof. Let î ∈ I be fixed. Without loss of generality we can relabel î = 1.
By contradiction, suppose Rn \ E1 is decomposable and let {Fj }j∈J be the family of its
indecomposable components given by Theorem 6.6.
It holds
∪
Rn \ E1 = (Rn \ E) ∪
Ei (mod Hn ),
i∈I,i̸=1
where, we recall,
(Rn \ E) ∪ {E
i }i∈I,i̸=1
is a family of indecomposable and pairwise disjoint sets.
87
Decomposition of BV functions
From the maximal indecomposability of {Fj }j∈J and {Ei }i∈I , it follows that
∃! ĵ ∈ J
s.t. Rn \ E ⊆ Fĵ (mod Hn )
and
∀j ∈ J, j ̸= ĵ, ∃! i ∈ I, i ̸= 1,
Fj = Ei (mod Hn ).
s.t.
We relabel ĵ = 1.
Moreover, we can found two sub-families {Eil }l∈L and {Eik }k∈K of {Ei }i∈I such that
{Ei }i∈I = {Eil }l∈L ∪ {Eik }k∈K ,
and
∪
F1 = (Rn \ E) ∪
Eil (mod Hn ),
l∈L
∀k ∈ K ∃!j ̸= 1 ∈ J
s.t. Eik = Fj (mod Hn ).
Observe that
Rn \ F1 = E1 ∪
∪
Eik (mod Hn ),
k∈K
where {E1 , Eik k ∈ K} is precisely the family of indecomposable sets given by Theorem 6.6.
Therefore
∑
P (Eik ).
P (Rn \ F1 ) = P (E1 ) +
k∈K
On the other hand
P (Rn \ E1 ) =
∑
P (Fj )
j∈J
=P (F1 ) +
∑
P (Eik ),
k∈K
thus
P (E1 ) = P (E1 ) + 2
∑
P (Eik ).
k∈K
This implies
∑
k∈K
i.e.
Rn
∑
P (Eik ) =
P (Fj ) = 0,
j∈J,j̸=1
Hn -negligible
\ E1 is equal to F1 , up to
sets.
Therefore Rn \ E1 must be indecomposable.
From this lemma, for every i ∈ I, Ei and Rn \Ei are indecomposable. Therefore the functions
χEi are BV (Rn ) and monotone, so that the decomposition of χE ,
∑
χE =
χEi ,
gives |DχE | =
i∈I
∑
i∈I
|DχEi | as required.
Step 0. We can assume without loss of generality that f ≥ 0: in the general case one can
decompose f + and f − separately.
88
6.2 The Decomposition Theorem for BV functions from Rn to R
Step 1. The sets E t := {f > t} are of finite perimeter for H1 -a.e. t ∈ R+ , thanks to the
hypothesis that f is BV (Rn ) and Coarea Formula. Therefore, Theorem 6.6 gives, for H1 -a.e.
t ∈ R+ , pairwise disjoint indecomposable sets {Eit }i∈It such that
)
(
∪
t
n
t
Ei = 0.
H E \
i∈It
In particular, the property of maximal indecomposability yields a natural partial order relation
between these sets: since t1 ≥ t2 gives E t1 ⊆ E t2 , it follows that, for H1 -a.e. t1 ≥ t2 ∈ R+ ,
∀i ∈ It1 ∃! i′ ∈ It2
s.t.
Eit1 ⊆ Eit′2 (mod Hn ).
Taken a countable dense subset {tj }j∈J of R+ , such that, for all j ∈ J, the sets E j := E tj
are of finite perimeter, the countable family {Eij }j∈J,i∈Itj can be equipped with the partial order
relation
′
′
Eij ≤ Eij′ ⇐⇒ tj ≤ tj ′ , Eij ⊇ Eij′ (mod Hn ).
Therefore there exists at least one maximal countable ordered sequence (here we do not need
the Axiom of Choice).
j
Let {Ei(j)
}j∈J one of these maximal countable ordered sequences.
Notice that, once one of these sequences is fixed, the index i is a function of j, by the uniqueness
of the decomposition {Eij }i∈Itj .
Step 2. Define
{
f˜(x) :=
∪
0
x∈
/
j
}
sup{tj | j ∈ J, x ∈ Ei(j)
otherwise
j∈J
j
Ei(j)
Clearly 0 ≤ f˜(x) ≤ f (x) for all x ∈ Rn . Indeed, the set
{
{
}
j }
tj | j ∈ J, x ∈ Ei(j)
⊆ tj | j ∈ J, x ∈ E j
∀x ∈ Rn ,
passing to the supremum one has f˜(x) ≤ f (x) for all x ∈ Rn . Moreover f ∈ L1loc (Rn ) and
0 ≤ f˜ ≤ f give f˜ ∈ L1loc (Rn ).
Step 3. Fix t ∈ R+ such that E t is a set of finite perimeter. Define Ẽ t := {f˜ > t} and let E t
i(t)
j
the indecomposable component of E t which is contained in a set Ei(j)
of the maximal countable
′
j
′
n
ordered sequence and contains another Ei(j
′ ) , for certain j, j ∈ J, up to H -negligible sets. This
is possible for H1 -a.e. t ∈ R+ .
Due to the maximal indecomposability property, one has that
′
j
j
t
n
Ei(j
′ ) ⊆ Ei(t) ⊆ Ei(j) (mod H )
∀tj ′ , tj ,
where tj ′ > t > tj .
t
Notice that, for H1 -a.e. t ∈ R+ , there exists only one of such an Ei(t)
among all the indecomt
posable sets Ei , i ∈ It .
t
We show that Ẽ t = Ei(t)
(mod Hn ), for H1 -a.e. t in R+ , in two steps.
89
Decomposition of BV functions
t
• First we show that Ẽ t ⊆ Ei(t)
(mod Hn ) for H1 -a.e t in R+ .
For x ∈ Ẽ t = {f˜ > t}, there exist j1 = j1 (x), j2 = j2 (x) such that
f˜(x) > tj1 > t > tj2
and
j1
j2
x ∈ Ei(j
∩ Ei(j
.
1)
2)
Since for all tj1 > t > tj2 it holds
j1
j2
t
Ei(j
⊆ Ei(t)
⊆ Ei(j
(mod Hn ),
1)
2)
t , hence
it follows that for Hn -a.e x ∈ Ẽ t x ∈ Ei(t)
t
Ẽ t ⊆ Ei(t)
(mod Hn ).
• Next we show the other inclusion up to countably many values of t.
′
t
t
Observe that set Ei(t)
is contained in Ẽ t for all t′ < t. In fact x ∈ Ei(t)
implies f (x) >
′
′
′
t > tj > t for some j ∈ J, hence f˜(x) ≥ tj > t . Thus for every tn ↗ t one has
∩
t′n ⊇ E t .
t′n <t Ẽ
i(t)
t \ Ẽ t ) > 0: from Ẽ t ⊆ E t
Suppose Hn (Ei(t)
i(t) it follows


(
)
∩
′
Ẽ tn \ Ẽ t  = Hn {f˜ ≥ t} \ Ẽ t
0 < Hn 
t′n <t
and this implies Hn ({f˜ = t}) > 0. This last condition can be satisfied only for a countable
number of t ∈ R+ .
t does not coincide with Ẽ t has zero n-dimensional Hausdorff
Therefore the set of t’s such that Ei(t)
t up to Hn -negligible sets. Since the
measure, i.e. for H1 -a.e. t ∈ R+ the sets Ẽ t coincide with Ei(t)
n
property of being indecomposable is invariant up to H -negligible sets, they are indecomposable.
In the following we will denote with t̃k , k ∈ K, the countable family of values such that
Hk := {f˜ = t̃k },
Hn (Hk ) > 0.
Step 4. The function f˜ is BV (Rn ) and has indecomposable super-level sets.
The indecomposability of the super-level sets of f˜ was proved in the previous step.
Using Coarea Formula, see for example Theorem 2.93 of [5], we get
∫ +∞
˜
|Df | =
P ({f˜ > t})dt
−∞
+∞
∫
=
≤
−∞
∫ +∞
t
P (Ei(t)
)dt
P (E t )dt
−∞
=|Df | < +∞.
Thus the function f˜ is BV (Rn ).
90
6.2 The Decomposition Theorem for BV functions from Rn to R
Step 5. Define the function fˆ := f − f˜. Clearly fˆ is BV (Rn ). The aim of the following steps is
to show that its total variation satisfies
|Dfˆ| = |Df | − |Df˜|.
Denote with E1t the super-level sets used to generate the function f˜: this can be done setting
i(t) = 1 for H1 -a.e. t ∈ R+ .
It has been proved that, for H1 -a.e. t ∈ R+ , one has {f˜ > t} = E1t , up to Hn -negligible sets,
therefore for such t’s
∑
P ({f > t}) =
P (Eit )
i∈It
∑
=
P (Eit ) + P ({f˜ > t}).
i∈It , i>1
We would like to show that, for H1 -a.e. t ∈ R+ , for every i ∈ It , i > 1, Eit is equal, up to
Hn -negligible sets, to one of the indecomposable components Êit̂ of {fˆ > t̂ }, where t̂ = t − t̃i
for a certain t̃i .
The index i in t̃i refers to the fact that its value varies with the indecomposable component Eit ,
i ∈ It , i > 1.
We prove it in the following three steps.
Step 6. Let t be such that the set E t is of finite perimeter and {Eit }i∈It are its indecomposable
components.
Let us prove that there exists a unique k ∈ K such that the set Eit , i ∈ It , i > 1, is contained
in Hk , up to Hn -negligible sets.
The set Eit is indecomposable and Eit ∩ E1t = ∅. Being E1j ⊆ E1t for all tj ≥ t, up to
Hn -negligible sets, it follows
(
)
Hn Eit ∩ E1j = 0 ∀tj ≥ t.
Therefore, from the definition of f˜, for Hn -a.e. x ∈ Eit one has f˜(x) ≤ t.
Again from the indecomposability of Eit and from the fact that Eit is contained in {f > tj }
for all tj ≤ t, it follows that there exists a unique l ∈ Itj such that,
Eit ⊆ Elj (mod Hn )
and
(
)
j
Hn Eit ∩ Em
= 0 ∀m ̸= l, m ∈ Itj ,
for all tj ≤ t.
(
)
′
If there exists a j ′ such that Hn Eit ∩ E1j = 0 then
(
)
Hn Eit ∩ E1j = 0,
∀tj , 0 ≤ tj ′ ≤ tj ≤ t
′′
on the other hand if there exists a j ′′ such that Eit ⊆ E1j , up to Hn -negligible sets, then
∀tj , 0 ≤ tj ≤ tj ′′
Eit ⊆ E1j (mod Hn ).
91
Decomposition of BV functions
Thus, being the definition
{
f˜(x) :=
∪
x∈
/ j∈J E1j
otherwise
0
sup{tj | j ∈ J, x ∈ E1j }
equivalent to
f˜(x) := inf{tj | j ∈ J, x ∈
/ E1j },
it follows that, up to Hn -negligible subsets of Eit , f˜ |Eit = constant, which belongs to {t̃k }k∈K .
t
In particular, we can order the sets Eit , i ∈ It , i > 1, as E(k,i)
where
{
}
t
{E(k,i)
| i ∈ Bkt } = Eit | i ∈ It , i > 1, Eit ⊆ Hk (mod Hn ) .
Note that Bkt could be empty for some t ∈ R+ , k ∈ K.
Step 7. Let t̂ > 0 such that the set Ê t̂ is of finite perimeter and {Êit̂ }i∈Iˆ are its indecomposable
t̂
components, for H1 -a.e. t ∈ R+ .
Let us prove that there exists a unique k ∈ K, such that the set Êit̂ is contained in Hk , up
to Hn -negligible sets.
Define
{
}
t̄ := sup 0, tj | j ∈ J, Êit̂ ⊆ E1j (mod Hn ) .
It follows that
f |Ê t̂ = fˆ|Ê t̂ + f˜|Ê t̂ > t̂ + t̄ > t̄.
i
i
i
For every tj in the countable dense sequence such that t̄ < tj < t̄ + t̂ there exists a unique
ī ∈ Itj such that
Êit̂ ⊆ Eīj (mod Hn ).
Due to the indecomposability of Êit̂ , and, for the definition of t̄, the index ī must be greater than
1.
Therefore f˜|Ê t̂ = t̄ and t̄ belongs to {t̃k }k∈K .
i
In particular, we can order the sets Ê t̂ , i ∈ Iˆ , as Ê t̂
where
i
t̂
(k,i)
{
}
t̂
{Ê(k,i)
| i ∈ B̂kt̂ } = Êit̂ | i ∈ Iˆt̂ , Êit̂ ⊆ Hk (mod Hn ) .
Note that B̂kt̂ could be empty for some t̂ ∈ R+ , k ∈ K.
Step 8. In this step we prove that, for H1 -a.e. t ∈ R+ , k ∈ K fixed,
t−t̃k
t
{E(k,i)
| i ∈ Bkt } = {Ê(k,i)
| i ∈ B̂kt−t̃k }.
Indeed, fix i ∈ Bkt
fˆ|E t
(k,i)
= f |E t
(k,i)
− f˜|E t
(k,i)
> t − t̃k .
Let us consider only the t’s such that the set {fˆ > t − t̃k } is of finite perimeter.
92
6.2 The Decomposition Theorem for BV functions from Rn to R
t−t̃k
t
For its indecomposability, E(k,i)
must be contained, up to Hn -negligible sets, in Ê(k,i
′ ) for a
unique i′ ∈ Iˆ .
t−t̃k
t−t̃k
Take then the set Ê(k,i
′):
f|
t−t̃
Ê(k,i′k)
= f˜|
t−t̃
Ê(k,i′k)
+ fˆ|
t−t̃
Ê(k,i′k)
> t̃k + t − t̃k = t.
t−t̃k
n
t
For its indecomposability, Ê(k,i
′ ) must be contained, up to H -negligible sets, in E(k,i′′ ) for a
t−t̃k
t
n
unique i′′ ∈ It , i′′ > 1. Thus i′′ = i and E(k,i)
= Ê(k,i
′ ) , up to H -negligible sets.
Hence
t−t̃k
t
{E(k,i)
| i ∈ Bkt } ⊆ {Ê(k,i)
| i ∈ B̂kt−t̃k }.
t−t̃k
t
The same argument, reversed, shows that, once i′ ∈ B̂kt−t̃k is fixed, Ê(k,i
′ ) = E(k,i) , up to
Hn -negligible sets, for a certain i ∈ Bkt . Hence
t−t̃k
t
{E(k,i)
| i ∈ Bkt } ⊇ {Ê(k,i)
| i ∈ B̂kt−t̃k }.
In an equivalent way, we can also say that, for H1 -a.e. t̂ ∈ R+ , k ∈ K fixed,
t̂+t̃k
t̂
{Ê(k,i)
| i ∈ B̂kt̂ } = {E(k,i)
| i ∈ Bkt̂+t̃k }.
t̂+t̃k
t̂
In the following we relabel Ê(k,i)
and E(k,i)
in order to have
t̂+t̃k
t̂
Ê(k,i)
= E(k,i)
(mod Hn ).
Step 9. Coarea Formula gives
∫ +∞
P ({f > t})dt
|Df | =
−∞
∫ +∞ ∑
∫
=
P (Eit )dt +
−∞ i∈I ,i>1
t
The final steps consist in showing that
∫ +∞ ∑
−∞ i∈I ,i>1
t
+∞
P ({f˜ > t})dt.
−∞
P (Eit )dt = |Dfˆ|.
(
)
Step 10. The set {t̃k | k ∈ K} is the countable set of values such that Hn {f˜ = t̃k } > 0 for all
k ∈ K.
Step 6 shows that, for H1 -a.e. t ∈ R+ and for all i ∈ It , i > 1, there exists a unique k ∈ K
such that f˜ |Eit = t̃k .
t
For every k ∈ K, let {E(k,i)
| i ∈ Bkt } be the set of indecomposable components of E t such
that f˜ | t = t̃k , i > 1.
E(k,i)
93
Decomposition of BV functions
Observe that
∑
i∈Bkt
t
P (E(k,i)
) are measurable functions of t, for all k ∈ K: indeed we have
D((f − t̃k )χH ) =
k
∫
∑
+∞
t̃k
∫
P ({f > t}i )dt
i∈It ,i>1
{f >t}i ⊆{f˜=t̃k }
∑
+∞
=
t̃k
P ({f > t}i )dt ≤ |Df |(Rn ) < +∞.
i∈Bkt
∑
Therefore the function t 7→ i∈B t P (Eit ) is integrable for all k ∈ K.
k
Using this notation, we can write
∫
+∞
∑
−∞ i∈I ,i>1
t
∫
P (Eit )dt
+∞
=
∑ ∑
−∞ k∈K
i∈Bkt
=
∑∫
k∈K
=
=
−∞
∑∫
k∈K
∫
=
t
P (E(k,i)
)dt
∑
+∞
P ({fˆ > t − t̃k }(k,i) )dt
t−t̃k
i∈B̂k
+∞
−∞
+∞
∑
i∈Bkt
−∞
∑∫
k∈K
+∞
t
P (E(k,i)
)dt
∑
P ({fˆ > t̂ }(k,i) )dt̂
i∈B̂kt̂
∑ ∑
−∞ k∈K
i∈B̂kt̂
P ({fˆ > t̂ }(k,i) )dt̂.
From Step 7 it holds
Ê t̂ =
∪
{Êit̂ | i ∈ Iˆt̂ }
i
=
∪ ∪
t̂
{Ê(k,i)
| f˜ |Ê t̂ = t̃k , i ∈ Iˆt̂ }
i
i k∈K
=
∪ ∪
t̂
{Ê(k,i)
| i ∈ B̂kt̂ },
k∈K i
we can write
∫
+∞
∑ ∑
−∞ k∈K
i∈B̂kt̂
∫
P ({fˆ > t̂ }(k,i) )dt̂ =
+∞ ∑
−∞
∫
+∞
=
−∞
94
P ({fˆ > t̂}i )dt̂
i∈Iˆt̂
P ({fˆ > t̂})dt̂ = |Dfˆ|.
6.2 The Decomposition Theorem for BV functions from Rn to R
Step 11. Finally we have
∫
|Df | =
+∞
P ({f > t})dt
−∞
∫ +∞
=
∫
P ({fˆ > t})dt +
−∞
+∞
P ({f˜ > t})dt
−∞
=|Dfˆ| + |Df˜|.
Since f has bounded variation we can iterate this process at most a countable number of times
generating the family of functions f˜l ∈ BV (Rn ), such that everyone of them has indecomposable
super-level sets, for H1 -a.e. t ∈ R+ .
Step 12. Let f˜ := f˜l be one of the functions generated in the previous steps.
If {f˜ < t} is indecomposable for H1 -a.e. t ∈ R+ , then f˜ is already monotone. Otherwise we
must again decompose f˜. If we succeed in decomposing f˜ in a countable sum of monotone BV
functions which preserves total variation we are done, since the decomposition of every function
of a countable family in a countable family gives at the end a countable family as required.
In that case define F̃ t := {f˜ < t} and let {F̃it }i∈It be the family of indecomposable sets given
by Theorem 6.6 for H1 -a.e. t in R+ .
As for the super-level sets, we equip the family {F̃ij }i∈Itj with the natural partial order relation
F̃ij ≤ F̃ij′
′
′
⇐⇒ tj ≥ tj ′ , F̃ij ⊇ F̃ij′ (mod Hn )
and call {F̃1j }j∈J one of the maximal countable ordered sequences.
Define
˜
f˜(x) := inf{tj | j ∈ J, x ∈ F̃1j }.
As in the previous case, one has that
˜
• f˜ is BV (Rn ),
˜
• {f˜ < t} = F̃1t up to Hn -negligible sets and for H1 -a.e. t ∈ R+ ,
ˆ
ˆ
˜
• define fˆ := f˜ − f˜ then fˆ is BV (Rn ) and
ˆ
˜
|Df˜| = |Dfˆ| + |Df˜|.
Recall that, for H1 -a.e. t ∈ R+ , {f˜ < t} is decomposable and Rn \ {f˜ < t} indecomposable.
∪
˜
Since {f˜ < t} = i∈It F̃it and {f˜ < t} = F̃1t up to Hn -negligible sets, Lemma 6.10 implies that
˜
˜
Rn \ {f˜ < t} is indecomposable, hence the super-level set {f˜ > t} is indecomposable for H1 -a.e.
˜
t ∈ R+ . Therefore f˜ is monotone as required.
Since f˜ has bounded variation we can iterate this process at most a countable number of
times generating the family of monotone functions fi ∈ BV (Rn ), which satisfies the theorem.
95
Decomposition of BV functions
Remark 6.11. Notice that in Step 10 we have also proved that
∑
fˆ|∪k∈K Hk =
f |Hk − t̃k .
k∈K
We show now another proof of Theorem 6.6, which uses a variational argument.
Proof of Theorem 6.5. This proof is divided into 4 steps.
Step 1. The previous proof shows that if {f > t} is indecomposable for H1 -a.e. t in R, then
the decomposition
∑
−f =
fi ,
i∈I
where {fi > t} is indecomposable for H1 -a.e. t in R, implies that also {−f
∑ i > t} is indecomposable. Hence, it is enough to show that there exists a decomposition f = i fi such that {fi > t}
is indecomposable for H1 -a.e. t in R.
Step 2. Let f ≥ 0 be a BV function and let Eit be an indecomposable component for the
super-level set E t = {f > t}. Consider the variational problem
{∫
}
inf
|u(x)|dx, u ≥ tχEit , |Du| + |D(f − u)| = |Df | .
Since |Du| ≤ |Df | < +∞, the above problem admits a minimum f1 , and this minimum satisfies
0 ≤ f1 ≤ t.
Step 3. Assume that for some t > t1 > 0 the level set E = {f1 > t1 } is decomposable: let E1
and E2 = E \ E1 be a decomposition such that Eit ⊆ E1 and
P (E) = P (E1 ) + P (E2 ),
P (E1 ), P (E2 ) > 0.
Define the truncated function
{
f˜1 (x) =
f1 (x) x ∈ Rn \ E2
t1
x ∈ E2 .
Clearly f˜1 ≥ tχEit and ∥f˜1 ∥L1 < ∥f1 ∥L1 . Moreover, since for t1 < t2 < t one has {f1 > t2 } ⊆
{f1 > t1 }, it follows from the indecomposability that
(
)
(
)
P ({f1 > t2 }) = P {f1 > t2 } ∩ E1 + P {f1 > t2 } ∩ E2
so that
f1 = f˜1 + (f1 − t1 )χE2 ,
|Df1 | = |Df˜1 | + |D((f1 − t1 )χE2 )|.
Hence if {f1 > t1 } is not indecomposable for 0 < t1 < t, the the function f1 can be decomposed
as the sum of two positive functions f˜1 , fˆ1 such that ∥f˜1 ∥L1 , ∥fˆ1 ∥L1 > 0 and
|Df1 | = |Df˜1 | + |Dfˆ1 |.
96
6.2 The Decomposition Theorem for BV functions from Rn to R
Step 4. The subadditivity of the norm and the previous step implies that
(
)
|Df˜1 | + |D(f − f˜1 )| = |Df˜1 | + |D f − (f˜1 + fˆ1 ) + fˆ1 |
≤ |Df˜1 | + |D(f − f1 )| + |Dfˆ1 |
= |Df1 | + |D(f − f1 )|,
and this with the fact that ∥f˜1 ∥L1 < ∥f1 ∥L1 yields a contradiction to the minimality of f1 .
Remark 6.12. In general the decomposition of f in BV monotone functions is not unique as
the following example shows.
f1
f2
f1
f2
(a)
(b)
f
(c)
Figure 6.1:
The function f in Figure 6.1(c) can be decomposed either in the way shown in Figure 6.1(a)
or in Figure 6.1(b).
In the simple case, where f is the characteristic function of a set of finite perimeter with an
indecomposable complementary set, there exists a unique subdivision of f as a countable sum
of BV monotone characteristic functions. Moreover in that case, due to the fact that the sets
Ei are pairwise disjoint, DχEi are mutually singular for all i ∈ I.
This property, which has been proved also for the decomposition of Lipschitz functions in
Theorem 6.2, can be false in the general case. As shown in the example below, one can have
97
Decomposition of BV functions
monotone BV functions, whose distributional derivatives are concentrated on sets with non
empty intersection.
Example 6.13. Let us consider a BV function f as in the Figure 6.2.
3
3
2
2
f
f1
1
f2
1
1
2
3
1
2
3
Figure 6.2:
In this case Theorem 6.5 gives two BV monotone functions f1 and f2 such that f = f1 + f2 .
Their distributional derivatives are
|Df1 | = 2δ0 − δ1 − δ3
and |Df2 | = 2δ2 − 2δ3 ,
where δx is the Dirac measure, δx (A) = 1 if x belongs to the set A, δx (A) = 0 otherwise. Clearly
these distributional derivatives are not mutually singular, since both have an atom in x = 3.
One can easily show that for any other monotone decomposition it is impossible to find two
disjoint sets on which the distributional derivatives are concentrated.
6.3
Counterexamples
As we have already said, the definition of monotone function could be given even for a function
which is only L1loc (Rn ). In that case one has to require that this function must have super-level
sets with finite perimeter, which is true H1 -a.e. t ∈ R for the super-level sets of a BV function.
The Jordan decomposition states that monotonicity is a sufficient condition for a function of
one variable to be of bounded variation. However, we cannot say that every monotone function
f : Rn → R defined as in Definition 6.4 is of bounded variation.
A counterexample is given below by a function, whose super-level sets are progressive configurations of the construction of a Koch snowflake.
Example 6.14. The Koch snowflake is a curve generated iteratively from a unitary triangle T
adding each time, on each edge, a smaller centered triangle with edges one third of the previous
edge, see Figure 6.3.
More precisely letting T0 be the equilateral triangle T with unitary edge, and Ti the successive
iterations of the curve, one has that at every stage
98
6.3 Counterexamples
Figure 6.3: Progressive configurations of the construction of a Koch snowflake
• the number of edges is Nk = 3 · 4k ,
( )k
• the length of the edges is Lk = 13 ,
• the perimeter of the iterated curve is P (Tk ) = 3 ·
( 4 )k
3
,
• the area of the iterated curve is
[
1∑
H (Tk ) = 1 +
3
k
2
j=1
( )j ] √
4
3
·
.
9
2
Denote with B the ball
B = {x ∈ R2 | ∥x∥ < R},
which contains the unitary triangle T centered in the origin: hence Ti ⊆ B for all i ∈ N.
Let Ek := B \ Tk for k ∈ N and define f : B → R in this way
f (x) :=
∑ ( 3 )k
k
4
χEk (x).
Clearly 0 ≤ f < 4, therefore f belongs to L1 (B) and Coarea Formula can be used to obtain its
variation.
Let us note which are the super-level sets and their perimeter:
• for t < 0 the set {f > t} = B and P (B, B) = 0,
• for t = 0 the set {f > t} = E0 and P (E0 , B) = 3,
• for 0 < t < 4 the set {f > t} = Ek̄ for the first k̄ such that
( )k̄
3 · 34 ,
99
∑k̄
k=0
( 3 )k
4
> t and P (Ek̄ , B) =
Decomposition of BV functions
• for t ≥ 4 the set {f > t} = ∅ and P (∅, B) = 0.
Thus this function is monotone and computing its variation one has
∫
|Df |(B) =
+∞
P ({f > t}, B)dt
−∞
∫ 4
=
=
P ({f > t}, B)dt
0
+∞
∑
k=0
( )k ( )k
3
4
·
= +∞
3·
3
4
which implies that f does not belong to BV (B).
In the case of Lipschitz functions Definition 6.1 and Definition 6.4 are equivalent.
Proposition 6.15. Let f : Rn → R be a Lipschitz function, then f is monotone in the sense of
Definition 6.1 if and only if f is monotone in the sense of Definition 6.4.
Proof. (⇒) Let f : Rn → R be a Lipschitz function which is monotone in the sense of Definition
6.1, then for all t in R the set {f = t} is connected.
We claim that {f > t} and {f < t} are open connected sets. Indeed, let us concentrate on
{f > t}, the other case is similar.
By contradiction suppose {f > t} disconnected, then {f > t} must have at least two connected
components. For t′ > t, such that t′ − t is sufficiently small, the set {f = t′ } is contained at
least in two of the connected components of {f > t}. Thus we have a connected set {f = t′ }
contained in two connected components of a disconnected set, absurd.
Since for H1 -a.e. t in R the sets {f > t} and {f < t} are of finite perimeter Proposition 2 in
[3] gives that the open and connected sets {f > t} and {f < t} are indecomposable for H1 -a.e.
t in R.
Therefore f is monotone in the sense of Definition 6.4.
(⇐) Let f : Rn → R be a Lipschitz function which is not monotone in the sense of Definition
6.1, then there exists a t in R such that the set {f = t} is disconnected.
For Theorem 6.1.23 in [25], every connected components of {f = t} coincides with a quasiconnected component of {f = t}, because {f = t} is compact.
This implies that there exists an open set G in Rn such that
∂G ∩ {f = t} = ∅,
G ∩ {f = t} ̸= ∅
and
(Rn \ G) ∩ {f = t} ̸= ∅.
From its continuity, f must be greater than t or lower than t over the all ∂G. Let us fix f |∂G < t.
The compactness of {f = t} gives the existence of a δ > 0 such that f |∂G ≤ t − δ. Thus, for
all ε ∈ (0, δ),
∂G ∩ {f > t − ε} = ∅ and {f = t} ⊆ {f > t − ε}.
100
6.3 Counterexamples
Therefore
G ∩ {f > t − ε} ̸= ∅,
(Rn \ G) ∩ {f > t − ε} ̸= ∅.
In addiction, defining L the Lipschitz constant of f ,
d({f ≥ t − ε}, ∂G) ≥
δ−ε
.
L
It follows that the open set {f > t − ε} can be decomposed into two open sets with positive
distance, in particular it is decomposable.
In the case
f |∂G > t,
one can similarly show that, for all ε in (0, δ), the set {f < t − ε} is decomposable. Therefore f
is not monotone in the sense of Definition 6.4.
The Decomposition Theorem for real valued BV functions of Rn is in some sense optimal.
Considering BV functions from R2 to R2 one can find counterexamples to this theorem, i.e.
BV functions which cannot be decomposed in sum of BV monotone functions preserving total
variation.
The crucial point is that we require to our decomposition, besides being the sum of BV
monotone functions, to preserve the the total variation, i.e.
∑
|Df | =
|Dfi |.
i∈I
Remark 6.16. For example, let us generalize as follows our definition of BV monotone function
to functions with values in a space of a greater dimension.
Definition 6.17. A function f : Rn → Rm , which belongs to [BV (Rn )]m , is said to be monotone
if the super-level sets
{f > t} := {x ∈ Rn | fi (x) > ti i = 1, ..., m}
and the sub-level sets
{f < t} := {x ∈ Rn | fi (x) < ti i = 1, ..., m},
are indecomposable, for Hm -a.e. t ∈ Rm .


f1
Let f : Rn → Rm a BV function f =  ... .
fm
For i = 1, ..., m, every fi is a BV function from Rn to R so that Theorem 6.5 applies.
∑ Therefore,
for every i = 1, ..., m, one has the decomposition in BV monotone functions fi = j∈Ji fij .
 
0
 ... 
 

Note that, if g : Rn → R is a BV monotone function, the function 
 g  is a BV monotone
 ... 
0
function too, from Rn to Rm , in the sense of Definition 6.17.
101
Decomposition of BV functions
It follows that we can decompose f in that way
 j


0
f1
∑ 0 
∑  ... 
  + ... +


f=
 ... 
 0 .
j∈J1
j∈Jm
j
0
fm
However, this decomposition does not preserve the total variation of f and one can only say
that




0
|Df1j |
∑  ... 
∑ 0 




|Df | ≤
 0 .
 ...  + ... +
j∈Jm
j∈J1
j
0
|Dfm
|
We give now a counterexample in the case of Lipschitz function from R2 to R2 . In this
situation we extend the Definition 6.1.
Definition 6.18. A function f : R2 → R2 , which belongs to [Lip(R2 )]2 , is said to be monotone
if the level sets {f = t} = {x ∈ R2 | f (x) = t} are connected for every t ∈ R2 .
We observe that if f : Rn → Rn Lipschitz is a monotone operator, then its level sets are
closed convex. Hence the requirement to preserve the connectedness of the level sets is weaker
than being a monotone operator.
From the Area Formula
∫
∫
H0 (f −1 (t))dH2 (t) =
det(∇f (x))dx
R2
R2
one can say that f −1 (t) is finite for H2 -a.e. t ∈ R2 , i.e. f −1 (t) = {x1 (t), ..., xq(t) (t)}. Therefore
there exists a measurable selection h : R2 → R2 such that h(t) ∈ f −1 (t) for all t ∈ R2 .
Note that the graph
G(f ) = {(x, f (x))| x ∈ R2 }
is closed, thus for Theorem 5.8.11 of [36],
∪
G(f ) = {(hi (t), t)| t ∈ R2 },
i∈I
where every hi is a Borel function and I a countable set.
Define, for every x ∈ Ai := hi (R2 ), the function fi (x) := h−1
i (x).
2
Being Ai the set where hi is invertible, fi : Ai → R is well defined and, in its domain, it is a
Lipschitz function with constant equal to the one of f . One also has f = fi in Ai .
Due to the injectivity of fi , for all t ∈ fi (R2 ) there exists a unique x ∈ Ai such that {fi = t} =
{x}, which is a connected set. Therefore, for every i ∈ I fi is a Lipschitz monotone function in
Ai .
∑
in sum of Lipschitz monotone
Thus, we can decompose f = i∈I fi . This decomposition
∑
functions fi preserves total variation as desired |Df | = i∈I |Dfi |. However, these functions
are not defined on the all R2 but only on the sets Ai ⊆ R2 for which we just know measurability.
102
6.3 Counterexamples
The fact that it is possible to extend these functions to R2 requires an additional property
1
of the function f . Clearly every fi can be
∪ extended to Ai preserving its Lipschitzianity .
2
Fix an i ∈ I. We have R \ Ai = j∈J Oj where the Oj are connected open sets. The
extension of fi on the all R2 must preserve monotonicity and the
∑ total variation of fi . For this
reason and due to the fact that we already know that |Df | = i∈I |Dfi |, the function fi must
be constant on the Oj with positive measure.
Therefore, to preserve the Lipschitzianity, fi must be constant on ∂Oj . Thus, for every j ∈ J
such that Oj has positive measure, there must be a tj for which H1 ({fi = tj }) > 0.
Note that, if for every j ∈ J the sets Oj have zero measure, the function fi is the only one in the
decomposition and is already monotone, therefore the only interesting case is when there exists
at least a j ∈ J where the corresponding set Oj has positive measure.
Thus one must have
H1 ({f = t̄ }) ≥ H1 ({fi = t̄ }) > 0
for at least a t̄ ∈ R2 . The condition H1 ({f = t̄ }) > 0 for at least a t̄ ∈ R2 is a necessary
condition for the decomposition of a function in that particular way.
Example 6.19. Taken a Lipschitz function f : R2 → R2 we have seen that a necessary condition
for its decomposition is
H1 ({f = t̄ }) > 0
for at least a t̄ ∈ R2 .
However, not all Lipschitz functions from R2 to R2 have this particular property. For example
consider
(
)
1 − cos( πx2 1 )
2
2
f : R → R , f (x) =
.
1 − cos( πx2 2 )
For this function the level sets {f = t} have zero length for every t ∈ R2 . Thus any
decomposition with the properties desired is impossible.
1
Thanks to Kirszbraun’s theorem, see Theorem 2.10.43 in [27], every fi can be extended to a Lipschitz function
of the all R2 . However, for our purpose, it is sufficient to consider the basic Lipschitz extension to the closure.
103
Acknowledgements
This thesis has been realized under the supervision of Stefano Bianchini. I am really grateful
to him for proposing me the topics of research presented here, for his precious help and his
enriching discussions made during these years of my PhD.
I would like to thank Giovanni Alberti and Gianluca Crippa for their useful suggestions on
the main topics of this thesis.
Deep gratitude is devoted to my parents and my sister, they always supported and encouraged
me with love.
Kindest thanks are to the friends that followed me from a distance and to the ones that I
had the pleasure to meet along these years in Trieste and that shared with me this beautiful
experience.
In SISSA I have met colleagues that little by little became friends. Since some of them are
like brothers and sisters to me I really need to thank this big family.
105
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